Wood carving is not something you pick up over night; it requires a lot of patience and practice. What you will find, however, is that as you practice you will develop intuition that will allow you to tackle increasingly difficult problems on your own. It's an enjoyable and relaxing activity that I would recommend to anyone interested in learning a new skill.
Step 1: Tools
You're going to need a knife. While you could conceivably use any kind of sharp knife, I highly recommend investing in something nice... I use the chip carving knife pictured here. High quality tools are going to be hand-made with tempered steel that will cut better and last longer than the low quality stuff. There are chip carving knives with different shapes that will make difficult cuts a little easier, although they aren't necessary for the beginner.
You'll be putting your knife/knives through a lot of stress and thus you will need to sharpen them. You can buy sharpening stones or you can use a combination of 600+ grit sandpaper and water or oil. Any piece of leather will work as a strop. I won't get into sharpening in this instructable, however there are plenty of resources out there for you to learn.
Step 2: Wood
However, it isn't crucial that you even know what kind of wood you have, so don't worry. Use what you have– that's part of the fun of carving wood.
Make sure the wood is dry; you'll have more control over it. Carving with wet wood adds a level of unpredictability, since as it dries it tends to warp and crack.
Step 3: Grain
You will want to start with straight-grain wood (which is exactly what it sounds like), where the grain runs in one direction. You can use the ends to determine what direction the grain runs in. The pictures below show you what to look for. Note that the grain can go down slightly... this will be important.
Basic vocabulary is as follows: when you are carving in the same direction as the grain, you are carving with the grain. If you are carving opposite the direction of the grain, you are carving against the grain. If you are carving perpendicular to the grain, you are carving across the grain.
Which way to carve:
You always want to carve with the grain or across the grain. The grain can also run up and down slightly, so make sure you are carving with the grain in the down direction. The next step contains pictures of carvings in each direction for your reference.
Step 4: Safety / Holding the Knife
The picture below will show you how to hold the knife. When right-handed, my left hand is holding the piece of wood and my right hand is holding the knife. My left thumb is on the back of the blade and will be providing the cutting force. Never push the blade forward with your right hand; you will not be able to move the blade with precision.
Notice that my fingers are out of the path of the blade.
Don't be silly:
Never push hard on the blade. If it gets stuck (because you carved too deep into the wood or the grain orientation changed), stop and backtrack. If you try to force the blade, it could slip and cut your finger.
There's no need to go fast... Speed doesn't help you carve better. Take your time, at least until you feel comfortable carving.
The sound and the feel of your cuts are important. You can hear the difference between carving with, against, and across the grain. This will be useful in developing intuition about the direction you are cutting in.
Step 5: Technique
You'll be making a scooping motion. First, the knife has to dig into the surface of the wood a little bit. The knife must then be pushed through the wood; it's all about the angle! As you push through the wood, angle the knife up ever-so-slightly and you will have shaved off a bit of wood. With many types of wood, the shavings will curl; this is an indication that you are going in the right direction.
You should remove the wood in very thin layers; if you go too deep, you'll end up tearing out on the way back up.
The pictures below will show you what it looks like when you go with the grain, against the grain, and across the grain.
Also, your shavings are fairly informative. Check out the difference between shavings going against the grain and shavings going with the grain.
Step 6: Practice
The more you carve, the more you will understand the grain. It's somewhat difficult to explain in a paragraph with a few pictures, but it will make sense as you experience it first-hand.
Start with gently curving objects that allow you some room to make mistakes. As you get better, add details regardless of the orientation of the grain; there is always a way to make the right cut.
Most important of all, have a good time! (and don't hurt yourself)