Learning to restring a guitar is a lot easier than you might first think, if you do it the right way. If you can't do it at the moment, and are taking it to your local store to get it done, now is the time to learn.
Do be careful though, there is a lot of tension on guitar strings, and if they come loose or snap, they can do you some damage (especially your eyes).
Also be careful to dispose of your old string carefully. Cats and dogs really like to chew on them, but they can go straight through their cheeks. So please dispose of them safely (perhaps rolled up and wrapped in paper?).
Step 1 - remove and replace
The first step is to totally remove the original string. This means taking any bits off the tuning peg, and taking out the "Ball" from the bridge end. There are two main types of electric guitar and two different types of bridge.
On a Fender type guitar (note not all Fenders, but most, and also many other brands) the string goes through the guitar and comes out the back of the guitar. You must check that the ball has been removed from here. If you do not you might get two that wedge themselves in there and it is a real task to get them out. You can check by holding the guitar up to the light and you should be able to see if it is still there. If you have any difficulties getting it out then try using the new fat 6th string to poke it out from the front of the guitar. You might also like to try removing the plastic back plate to make it easier to see what you are doing. I leave my back plate off my strats all the time to make string changing a load quicker.
On a Gibson type guitar (note again that it's not all Gibsons) The balls are just hooked through the bridge. You can see these quite easily and should be able to remove it without any problem.
Once this has been done you feed the string into the hole where the ball was. On Gibson guitars this means just poking it through the hole toward the neck (sharp end first) and pulling it through until stopped by the ball. On Fender types you must put the string (sharp end first) into the appropriate hole in the back of the guitar (under the plastic rear plate), push it through, grab it at the front and pull it all the way through until stopped by the ball.
Step 2 - winding on the peg
This part is the same for all types of guitars (except classical guitars). The most important part of this is getting the string on the right side of the peg, and here is how to do it. First of all line up the hole in the peg so it is facing straight down the neck. Put the string through the hole and pull it back so you have some slack. The amount of slack you need will vary, depending on the thickness of the string. The 6th string only needs about 7cm but the 1st string can take 10cm, or thereabouts. Now hold the string in place with your right hand, just hold the peg so the string cannot slip and then turn the peg ANTI-CLOCKWISE if you are looking at the peg (if the peg is underneath like on some Gibson guitars then it looks clockwise from above).
As you continue turning the peg the string will wrap around it. The first time round the string should go above the hole (and the slack poking out) and above the string and after 1 lap it should go under the string. This will make it lock onto the string as it gets tighter (it is OK for all the wraps to go under, it is just more secure if you get the lock). You should aim for a least 3 wraps on the 6th string and 5 wraps for the 1st string. More will not hurt, but less and the string may start to slip. Try not let the string overlap itself, as this may make it easier to break.
Step 3- Tuning up
The next step is to tune the string. I strongly recommend getting an electric tuner, as it is important to hear what the guitar should sound like when you are learning, and electric tuners are pretty cheap these days (I recommend the Boss TU-12 Tuner. I have had mine for 10 years, it gets a beating, but is still working fine). If you have one then tune up using this, if not, tune to another string that you know is in tune. If this is not an option, then tune up to a keyboard, pitch pipes, or whatever is available. See the lesson on tuning in The Basics.
Step 4- Stretching in
And lastly, and quite often forgotten, is to stretch the string in. Just gently pull on the string with your right hand, using your left hand to hold the string in its correct position in the nut. You should notice it going out of tune considerably, and will need to tune it again. Continue stretching until you no longer need to tune it up.
Locking pegs (like those found on many Stratocasters) speed up string changes considerably. With this type of nut, the process of string changing is the same but it is not important to get lots of winds on the peg. Just feed the string through the hole in the peg, tighten the wheel on the back, and tune up. I have this system on my red Stratocaster and I think it is great. Never had a problem with them, ever. Highly recommended.
String Locking Systems
The other type is patentented by Floyd Rose, but are commonly known as Locking tremolo systems (found on most Ibanez type rock guitars and most metal style guitars). The idea of this system is to lock the string using small clamps so it cannot slip and go out of tune, even when the whammy bar is being thrashed. Tuning these is quite a skill and will have to be another lesson unto itself. The system is good, and stays in tune, but also has it's flaws (like if you break a string then all the strings go out of tune, and, they are quite hard to get in tune. For more information try your local store and ask for help. I do not recommend this type of system for beginners at all. Use these only if you know what you are doing or they are a complete pain in the butt.
Video Part 1
Video Part 2