Stringing a guitar is a very controversial subject. Everyone has their own tried and true methods for keeping their guitars in tune. There are also many chapters in many books going into great detail on this very subject. What I'm going to attempt here is to explain in it's simplest form, how to string a basic electric guitar.

First of all I'm going to assume your guitar is clean and adjusted properly. (There's no point in putting strings on a dirty guitar with a bent neck and intonation problems). Next I'm going to assume that you've also made sure your hands are clean and free of oils. (Nothing kills that "new" sound faster than oils eating away at your strings).

Now for the actual stringing. Like I said though, this is just the basics: It's aimed at standard electric guitars. This is aimed at guitars with stock, non-locking tuners, nothing else. No debates over whether to replace one string at a time or remove all the strings at once and start from scratch. Nothing discussed about quality or condition of tuning keys, the shape of the nut and bridge saddle(s) etc....

Removing the old strings

To remove the old strings you can either unwind them off the posts* or slacken them fully before taking a wire cutter to em. With acoustic guitars, the strings are held onto the bridge by a pin. So, you also have to remove the pin carefully with a pin puller or a pair of pliers. If the pin is stuck, try putting your hand in the sound hole and pushing up from inside the guitar. As a last resort, you can leverage the pin out with a screwdriver but for safety sake, put down a piece of padding so that you don't damage your guitar. Dispose of your old strings. If they're still in one piece you can keep em as emergency extras but personally I'd just toss em out cause used strings are kinda gucky.

*If you have a string winder, great. If not, this is the first of several times in this process that you will wish you had one.

Replacing with new ones

Starting with any string you want, thread the new string through the base of the guitar at the bridge. Thread either from the rear of the guitar through the body on a strat type, or through the bridge attached to the front of the guitar on a les paul type. With acoustics, put the ball of the string into the hole and replace the pin so that it holds the string tightly in place. Be careful that the middle part of the string doesn't get crimped or bent when you are handling it...

Article continues, with diagrams:
http://www.guitartricks.com/beginners/beginstring.shtml