Training animals is not just restricted to dogs, horses and random circus animals, in fact any animal can in theory be trained, as long as it has a longing for a treat (edible as well as any other kind of treat. Some use toys). You can even clicker train a gold fish!

You will need:

A cat (duh!)
Irresistible treats
Patience
A clicker

Why a clicker, you may ask? And what is it anyway?

A clicker is basically just a device that does one simple thing - it makes a clicking noise. They are pretty inexpensive (you can find then cheap on ebay) and comes in various shapes and sizes to fit you. But they all do the same thing, that is clicking.
Its not that training an animal is completely impossible without a clicker, but a clicker does provide some benefits in training. Normally you would use your voice to praise the pet for doing the right thing, but your voice and/or wording isn't always consistent. A clicking sound is! Having a consistent sound that is always associated with positive reinforcement will aid the learning process.

What can you teach a cat?
Basically you can teach a cat pretty much the same tricks as you could teach a dog, and in much the same ways. If you're going to look more into clicker training of cats, don't necessarily restrict yourself to only cat related articles or books, because there aren't exactly that many out there. But for dogs on the other hand, there is tons!
The main difference between training a dog versus training a cat, is most likely the social aspect. A dog sees you as a part of its pack, and probably also as its leader, and that means a lot when it comes to training. A cat on the other hand, probably just sees you as a big clumsy food provider, so cats will most likely not work for you, as much as they would work to get a tasty treat!

Step 1: Introducing the Clicker

The first thing you need to teach your cat, is that the clicking sounds equals treats, and also that the clicking sound isn't dangerous.

It's pretty simple. Click and give a treat. Repeat. Preferably click first, then give the treat right after, or somewhat simultaneously while the cat is eating the treat. But clicking after the treat has been given and eaten is too late for the cat to be able to make a proper connection. For most cats, this doesn't take long for them to figure out, and you only need to do this a few times.
If you want to test to see if the cat has figured out the connection, try clicking once and see if it reacts. If it expects a treat, it got it figured out and you can move on to the next step. If not, repeat this step a few more times.

Step 2: Teach Him His Name

I've once read somewhere that the first trick you teach a pet, is not to be something bodily, as in something it can do without you specifically asking for it. Although I don't have any test subjects to prove this claim, I can vouch that it has caused some problems that my cats first trick was giving a high five. Every time he gets confused (and he does that a lot), he always resorts to waving his paws around in desperation, although the trick he is supposed to do isn't one where he is going to use his paws. However, it has opened some benefits regarding teaching him other tricks, like holding a ball.

It's up to you where you start, but teaching your cat its name is a very good and easy beginning. Theories claim that a cat can better recognize its name if it has an "I" sound in it, like Mittens, Pussy, Tiger and so on. If your cat doesn't seem to understand its name, you can always use a calling sound.

Get started:
Sit next to your cat and say his name, then give a treat and click. Do this a few times.
Say his name again, but this time wait for an reaction. If he is too excited about the treats and already have your full attention, wait a bit for him to calm down and looking somewhere else. Say his name again, and see if he reacts, preferably by looking at you. If he does, give him a treat and click. If he doesn't, repeat the beginning of the step a few more times.

Teach him to come:
Once the cat seems to know that his name (or a calling sound) means treats and will react when you say it, move away bit so the cat has to walk a few steps to get his treat. Say his name and let him come pick up his treat, then click.
Gradually extend the distance between you and your cat each time. However, its important that he doesn't just follow you once you moved away (it would be tempting, you have the treats after all!), because he is only supposed to move towards you once you call his name, otherwise he won't get the connection of 'name=come'.
If you eventually manage to move far away from the cat and only have him follow you once you call his name, try challenge him a bit more and call from the nearest room, so he can hear you, but not see you.

Depending on your cat, you might not want to teach one trick in an entire sitting. If the cat seems distracted or just not interested, take smaller steps over longer time.

Step 3: Sit!

A classic dog trick. Sit on your bum! This trick can be pretty practical in many situations, and is a good way to start your cat talent show, to then follow it up with all the other neat tricks you teach your cat. And its also adorable!

This method of training can be used in many cases when teaching your pet a trick. I personally call it the "accidentally did the right thing" method, and as you can probably tell, its not entirely reliable. But for simple stuff like sitting, its good enough. It all boils down to the point, that the cat will eventually do the right thing if you're patient enough.

However, you don't just sit and wait for it to happen by accident, because that could take a while. You can encourage it a bit!
Take a treat and hold it next to your cats nose, so he knows you have something - but don't give it to him. Hold the treat over his head and move the hand a little back. If he decides to follow the treat floating over his head, he will most likely sit on his butt to be able to keep track of it. Once he does that, click and give the treat. Do this a few times, but remember to say "Sit" (or whatever command you choose) while you are holding the treat over his head, making him sit down.
Please don't force your cat to sit by pressing down on his rump. It's okay to give a gentle pad or push if he has already learnt most of the trick already, but forcing the animal in a specific position like that against its will, will only result in a negative association with the trick, and will not benefit you at all - quite the opposite!

Obviously its important that the cat is already standing before you ask him to sit, but even if he sits before you hold his hand over his head, but after you said sit, click, treat and praise for a good job! He might not do it because of the command at this stage, but as long as he learns a connection between the command and his own action, he will eventually figure it out.
Remember not to hold the treat too far over his head, or he might just jump up after it. If he jumps, then sits, and you click for that behaviour, he will most likely develop 'pigeon superstition' and jump up every single time you ask him to sit. If this is not part of the trick, only click and treat if he sits without jumping first. If he has trouble figuring out how to sit, without jumping (or something else) first, you can click and treat the first few times, and then eventually 'shape' him away from the unwanted part of the trick. We'll get back on shaping in the next step.

If you teach him to sit mostly by holding your hand with the treat over his head, you might eventually shape that into a hand gesture. When I tell my cat to sit, I say "Sit" and point a hand or finger over his head (without a treat in hand) in a stroking gesture, even if I'm standing up and he is down on the floor.
It is possible to teach the cat to only do it by verbal command, but it takes more time and effort. For that, you will need to eventually stop with the hand gesture, by slowly reducing it and using the verbal command more than the gesture, and eventually only click and treat if he is able to sit without the hand gesture.
However, as with people, cats are different in regards to learning. My cat doesn't respond well to verbal commands alone, and I always need to combine it with a hand gesture, or only use hand gestures. Work with what you have, and pay attention to what kind of student your cat is, and how he learns best.

Step 4: Walk in a Circle (Shaping)

Shaping is a very big part of training animals in general. In fact, you can say almost all tricks revolves around shaping in one way or another. To shape, in regards of animal training, means slowly encouraging the animal in the right direction of what you want it to do, one session at a time. Obviously you can't just tell a cat "Walk in a circle" and then he'll do it, you first need to show him how to walk in a circle, then slowly incorporate a command or hand gesture while doing so.

Get started:
I initially learnt about this trick from a rat training video. The steps are pretty much the same, which just goes to show that you don't need to restrict yourself with "cat tricks", as there is no such thing. All mammals can more or less learn the same things, regardless of specie.

Put a treat in your hand and let the cat know you have it (make it take a sniff, but not be able to eat it). Circle your hand around the cat, forcing it to spin around itself to follow the treat. If you want a verbal command, say this word while its spinning. Click and treat.
The circle might not be perfect in the beginning, but thats okay. As long as you slowly strive for it to be better and better for each time you do so.
Once you are satisfied that the cat knows how to follow your hand in a perfect circle, move your hand up a bit, meaning you will now stop leading the cat with the treat, as much as you are merely guiding it in the right direction. Click and treat. Repeat a few times until the cat is ready for the next step. If the cat is not ready for the next step, go back and repeat a few more times.
For each session/step, move your hand higher above the cats head, until you eventually is just drawing a circle with your hand over its head.

Step 5: Doorbell

This is probably one of the most practical tricks to teach a cat. If you have an outdoor cat, you probably know how annoying it can be when your cat calls for attention to let you know he wants to go outside. This attempt to get your attention can be anything from loud meowing, scratching the furniture, or scratching the door. Anything that will eventually make you let it out. In fact, there are many examples of cats training their humans to get what they want. For example, if the cat always gets kicked out of the house as a "punishment" every time it scratches the sofa, it will soon find a connection between scratching sofa=going outside, and suddenly ruining the new sofa becomes the cats main way of communicating that it wants to go out.

So wouldn't it be neat if YOU got to decide how the cat tells you it wants to go out, rather than the other way around? So much furniture can be saved on that account!

Get started:
Get yourself a bell of some kind. Its not so important what it looks like, as long as you're able to hear it from wherever you usually are in the house, when the cats wants to go out. Otherwise it won't be of much use. A traditional bell is probably the best to use, as its easy for the cat to use and provides more sound with less effort. If you're really tech savy, you could even get an electronic doorbell as well, as long as you're sure the cat is able to push the button with ease.

Set up your bell in cat friendly hight near the door where the cat is usually let out. Introduce the cat to the bell by ding'ing it. Give it a treat (but don't click) the first few times where you ding the bell and the cat pays attention to what you're doing.
Now, I always had the advantage of having a cat that always turns to using his paws when he is exited, so I could use that to my advantage. If you're cat usually don't use its paws much, using its nose is just as good. The main thing here is to teach it how to operate the bell on its own.
You might want to guide the cat by lifting its paw, and using the paw to ding the bell. Click and treat. If the cat finds it too uncomfortable that you're forcing its paws like that, try something else. If paws is not an option, guide the nose with a treat in your hand towards the bell, and ding it on behalf of the cat. Give the treat. If the cat nears its face towards the bell on its own, even if it doesn't ding it, click and treat. Same goes for paws. If it makes an attempt to ding the bell on its own (even if its far off), click and treat even if it doesn't make a sound. Eventually it will figure out that it needs to do something with the bell. Once it tried long enough and managed to give the bell a few dings, only click and treat if the bell makes a sound. Repeat this until the cat got the connection between making the bell go ding, and getting a treat.

Once the cat has fully understood how to operate the bell, stop giving it treats, but move the cat outside. Do this every time the cat uses the bell, even though it will in the beginning only use the bell for treats, it will eventually get a new connection to the bell, that is going outside.
If the cat should stop using the bell, go back a step and keep giving treats when it uses the bell, then move it outside. Eventually only open the door when it uses the bell.

Step 6: Target Stick

Using a target stick can be beneficial in many other ways, and might even help teach the cat new tricks.
Teaching a cat to use a target stick is really simple! The concept is simply having the cat follow a stick.

Get started:
Find a stick of any kind. I use a telescope antennae from an old radio, because it can be folded together when its put away.
It can be a good idea to have some kind of 'thing' on the end of it, maybe bright coloured or so. It will make it easier for the cat to recognize the tip of the stick, but it will also work fine if you don't have anything at all. That 'thing' can either be a tiny ball or some rubber bands tired around it. A telescope antennae usually have a little lump already, which is sufficient.

Introduce the stick to the cat. Most cats are curious by nature and probably want to sniff the stick to inspect. Click and treat when the cats face nears the tip of the stick. Repeat this a few times.
Increase the length between the cat and the stick a bit, so he has to stretch a bit to reach it. Click and treat when he approaches the stick. Gradually increase the length more and more until he eventually has to walk a few steps to reach it. If he doesn't move, go back a step and repeat it a bit more.

Once he gets the hang of it, increase the length with a meter or a half, and have him walk around in a pattern you "draw" with the stick. Don't make it too long, or he might get impatient. Practise this some times and slowly increase the length he has to follow, and challenge him with walking a certain path. Don't challenge him more than hes patient for. Eventually have him follow the stick to jump on a chair, walk up some stairs, or even jump over an obstacle. Click and treat as usual.

Step 7: Knowing When to Stop

Cats, like any other animal, has limited patience. If you try too hard, or the cat just gets confused, don't get frustrated with him. It will only make it worse.

Cats are different individuals, and its hard to say exactly for how long you can expect a cat to concentrate on the task, but somewhere between 5-10 minutes at a time, some maybe less, some maybe more. If you keep pushing your cat even though it lost interest or is too confused about the task, the cat will have a negative experience regarding the training, and any future training might be even harder to do. Respect your cats limits, and take a break when it needs to.

If he looses interest or starts grooming himself, this is a sign that you need to stop, or at least take a break for an hour or so. A confused and frustrated cat will often begin grooming itself, often in a vigorous way. If your cat refuses to break from his grooming, the sign is clear - let him be.

Step 8: Let It Be a Positive Experience!

Let it be something fun, for you and your cat! If things start to get frustrating, it might be time to stop.
Training your cat is a great way to bond with your pet, and a fun way to keep your cat activated and entertained, and make him work for his treats.

Remember its all about positive reinforcement! If there is a reward, there is motivation. Don't punish or get mad if your pet doesn't do as you want, because it will only make the experience unpleasant for the pet, and make it even harder to train him. Your pet isn't difficult because he wants to annoy you, he is most likely not doing things right plain and simple because he does not understand what you want - help him understand, instead for getting mad.

There are tons of other tricks you can teach your cat! Go search on the internet, or maybe go to the library and look for books on animal training. You might find a lot of fun tricks meant for dogs, but cats are just as likely to learn the same tricks!

Introduction: How to Clicker Train a Cat