Repairing a bike inner tube is not only simple and cheap, but it's also mandatory in respect for the environment. Indeed a punctured tube can be repaired many times before being thrown out. A different matter is when the tube is cut or the valve leaks, since it's more difficult repair the tube. Anyway most of the times the leak is caused by a nail or a thorn, and the hole is very tiny.
Although many of you already know this process, I'll explain how to easy add a patch to a pierced bike inner tube.

Step 1: The Necessary

All you need is a pump and a repairing kit. I take for granted you already know how to remove the wheel from the bike and the tube from the wheel, so I don't deal this topic here, ask in comments if you fall in some obstacle.
The repairing kit is composed by some patches (probably of different sizes), some scratch tools (sandpaper or a little metal grater), a little tube of mastic (a type of glue) and maybe some little rubber tubes to spread mastic.

Step 2: Inflate

First thing to do: find the hole. This could be simple or pretty long. If you're lucky the hole is big enough to listening the air flowing out when you inflate the tube. But many times the hole is so tiny that air comes out very slow and it's difficult to find, in this case go to next step.

Step 3: Blblblbbl

To find the tiny holes inflate the tube. Don't be scared to inflate the tube very much, it could also become double in dimensions without exploding, and this will help you find the hole, since some hole remain closed with low pressure. At this time dip the tube into water, a sector at a time, until you see the bubbles coming out.

Step 4: Mark

Mark the hole with a permanent marker, so you will not have to make the entire process a second time if you look away for an instant ;-)
Make the mark twice as big as the patch with an "X" through the center of the hole, it allows for spreading out glue if too much is applied.

Step 5: Scratch

With the help of sandpaper or grater scratch the rubber all around the hole, this will help the mastic to adhere to the tube. No reason to wash away the rubber powder when finished, just blow it away or clean with your hand.

Step 6: Mastic

Time to spread the right amount of mastic on a surface a little wider than the patch. Move the head of the mastic container from center to the border, as pizza maker does with the tomato sauce.
Use a thin layer of glue. More isn't better and if it is cold will take much longer to cure. If too much comes out use the small rubber tubes in the repairing kit to spread it out.
The mastic will now dry in a few minutes. About 10 minutes will be probably a good time to wait before attaching the patch.
Don't touch the area that has been roughed up. Oil from your hand will contaminate the surface and impact glue adhesion.

Step 7: Patch

The patches have a metal cover on the glue side, and a plastic foil on the other one. When you remove the metal cover, the plastic transparent sheet should remain in place, you'll remove it after the gluing process.
Push the patch very hard over the dry mastic, you should use now an hard tool to make pressure, as the back of the pump. The more pressure you apply the more efficient the fixing will be.

Step 8: Strip

To remove the thin plastic sheet you have to keep down the patch red border with your nail. It's not a big problem if the border detaches, anyway try to leave it glued. If the plastic layer won't release from the top of the patch, leave it be, when 60+ psi push it against the inside of the tire, it isn't going to make any difference.
Now you could in theory assemble the tube on the wheel, inflate it and ride. So don't be afraid to do that, BUT...
Remember to check both the inside of the tire and the tube protector strip on the rim for what caused the hole! If you miss that step you'll probably incur in the same identical accident for the same reasons.

Step 9: Inflate Again

Otherwise this is probably your reserve tube, you are at home fixing it meanwhile you relax your limbs after an hard riding morning, and you can wait some more time before inflating the tube to check the good air detaining. Let's say ten minutes...
Inflate a lot the tube and hang it somewhere in your haven, if it's still inflated the next day your reparation is perfect, if not restart from step two ;-) Obviously you can test the tube into water right away.

Step 10: Deflate and Roll Up

Now deflate (this is the tedious part), roll up the tube and hold it with an elastic. If you predict to not use it in a few time, you probably should scatter it with talcum powder, so the rubber will last much more time.
Also pay attention keeping your reserve tubes in the little bag under the saddle, because continual rubbing can wear out the borders. A solution is to put the tube in an old sock, which makes a great rag and you can put it on your hand if you need to swing a derailleur out of the way to remove the rear wheel.
And remember, always bring at least a reserve inner tube with you, among with the necessary to change it, and to be safe bring also a repairing kit, it happens to get three or more blowouts in the same trip! I'm not speaking about city cycling ;-)