Admit it... your golf club grips have got to get replaced this year -- the same thing you've been saying now for the past three seasons. But, you know the local pro shop charges $5.00 per, plus the cost of the grips, and you're sure you could do it yourself if you just had a little direction. You've come to the right spot, because what follows is a tutorial on replacing your own golf grips (and maybe a buddy or two that also need it done). And you're right; you can do it yourself.
Evaluate
The first thing to do when faced with a re-grip job is to evaluate what is already there. Are they rubber? Cord? Half-Cord? Leather? Next, take a moment to measure them. Yes, believe it or not, grips come in different sizes for both the core (inside) and diameter (outside) or can be adjusted to different sizes, and a proper fitting grip is very important to hitting consistent golf shots. There are many tools available from various supply houses like Golfsmith or Golfworks.com, and I'll mention them along the way.
The first is a grip gauge -- a useful and inexpensive tool to have.
Measure the grip approximately 2" down from the butt to determine its size.
Take note of the size, as you'll need it later on (in this example, the grip is + 1/32" over standard).
Removal
The next step is removal of the old grips. This step can be as simple as slitting the grip lengthwise... 
(Use a utility knife with a reverse blade...)
And then peeling it off (this is mostly what you'll get):
Or it can be as difficult as chipping the grip off with a curved chisel in tiny, frustrating, curse-inducing increments (rare, but it happens).
Generally speaking, if the grip is 2-3 years old, it will come off with relative ease. Older than that, and the rubber starts to break down and seems to nearly fuse to the shaft of the club. Once the grip itself is removed, scrape and/or peel the excess grip tape from the shaft to get it to bare metal or graphite. The last bit of residue can be removed with a solvent like acetone or paint thinner. The purpose of getting the shaft clean is to ensure a smooth surface for the new grip, as well as a clean surface for the new grip tape and solvent to adhere to.
While this is rare, I'll mention it here -- if any adjustment is to be made to the shaft length (addition or subtraction) now is the time to do that. I recently re-gripped an older driver and added a half inch of length to try and squeeze a few more yards out of it -- and this was the process used to get ready for that addition.
Often times, the selection of a new grip is made before you have the complete information on the butt diameter of the existing shafts. While there is certainly a better fit by using the proper core diameter on the corresponding butt diameter, most rubber grips will fit either way (up or down). Just keep in mind this is OK to do most of the time, but it won't always work.
Preparation
Once the shaft(s) are clean, your next step is preparation. Using your grip gauge, measure the butt diameter of the bare shaft (again, appx. 2" down from the end) and make a note of it. This shaft has a .600" diameter:
Butt diameters generally fall into two sizes: .580" and .600". Conveniently enough, the core diameter of most grips are also found in these two sizes. Many grips are available in both sizes so, regardless of the butt diameter, a good fit can be achieved. Refer back to the original grip measurement taken at the outset. By comparing this measurement to the standards (found right on the grip gauge) you can determine if the grip you need can be purchased as 'standard' size, 'mid-size', or 'oversize,' or if the shaft needs to be "built up" with a layer(s) of masking tape.
Based on your hand size (see chart), if it is necessary or desirable to create a larger grip than you can purchase ready-made, you can do so by applying a single (or multiple) wrap of masking tape to the butt end of the shaft.
This method can also be used to address a specific fit issue -- for example, a right-handed person who plays golf left-handed may have need for an enlarged grip only where the right hand grasps the club (their right -- dominant -- hand is larger and thus benefits from this fitting method).
When applying the build-up and/or double-sided tape, simply hold the shaft of the club between your legs with the butt end in front of you. While it may seem a bit archaic, it's easy, fast and works great. Simply spin the shaft with one hand while spiraling on the tape with the other. A little trial and error is necessary to learn the angle at which to apply the tape, but after a couple of false starts, you'll get the hang of it.
Now that you have all the dimensions to either duplicate, or create, the correct grip size, the replacement grip can be selected. I don't have the exact number, but would estimate that grip options range well into the hundreds these days. The primary categories of grips are: Rubber, Cord, Half-cord, and Composite (polymer). Within each of these categories lay a plethora of choices for style, color, actual material, weather condition suitability, and on and on. Since the categories and sub-categories of grips is so vast, I will not attempt to address them all here. Rather, you will soon find another complete article discussing in more detail these grip options. The grip I've selected here is a Men's .600" diameter Round. This is coded just inside the mouth of the grip as: M60R.
Once the proper base diameter has been established, you will need to apply a spiral of double-sided adhesive tape (generally 3/4" width).
Double-sided tape is also available in pre-cut lengths of 1.5" wide tape. Either method is fine, but my experience has shown that spiraling the narrower tape is easier, faster, and yields a perfectly adequate adhesion of the grip.
When applying the double-sided tape, leave about 1.5 - 2" of excess tape available at the outset.
You'll need this to wrap over the open end of the shaft before the grip goes on.
Be sure to peel the protective layer off of the double-sided tape.
Installation
Now that the handle of the shaft is completely prepared, you are ready for installation. You'll need some kind of grip solvent to activate the grip tape and allow the grip to be easily slid on to the shaft. Again, grip solvents are a readily available from golf supply houses and you can even use acetone or lighter fluid in a pinch (use extreme caution, as these materials are flammable!). It is highly advisable to use a low CFC and non-flammable grip solvent.
Place the shaft in a rubber shaft clamp.
Then, clamp that into your bench vise. 
(Note: for a club-maker/repairman, a bench vise is a must-have item. If you don't have access to one, you'll simply have to enlist the aid of a buddy to hold the club for you while installing the grip. On second thought, it's probably easier to go out and buy a bench vise).
Take note of the club head position as you want to have it as near to the final installation orientation as possible.
Once the shaft is in place, use some kind of device to catch or capture the solvent spill. A paint roller pan works well or any large, shallow basin that won't be harmed by the solvent (glass, metal, or sturdy plastic are all OK).
When you have readied all of this, its now time to prep the grip and actually install it on the club. For most rubber grips, just applying a liberal amount of solvent in the grip, covering both ends.
Don't forget the hole in the butt end: 
Giving it a shake to distribute the solvent evenly is all you need. 
For stiff cord grips or especially the polymer grips (Winn, Karakal, etc.), place the grip in a microwave oven for 6-8 seconds before applying the solvent. Sounds weird, I know, but it really does help with this type of grip as they otherwise can be a bit of a challenge until you have some practice.
When you've wetted the inside of the grip with solvent, pour the excess solvent over the taped portion of the shaft, wetting the double-sided tape as well.
Then, quickly -- but without rushing -- lightly squeeze the open end of the grip to create a slightly oval shape, place this end over the club shaft... 
... and in one smooth, continuous motion, using both hands, slide the grip onto the shaft until is fully seated against the butt end. 
This process can slightly elongate a rubber grip, so to remedy this, place one hand against the butt of the shaft to hold the grip in place and gently pull back on the other end of the grip to gather it back to its original shape.
Alignment
At this point, there is but one thing left to do, and that is alignment. Again, moving quickly, but without rushing, remove the entire club from the bench vise and remove the grip clamp as well. Place the club on the ground and, while sighting down the shaft, make any final adjustments to the left or right to align the grip with the leading edge of the club face.
Though most grips are round, and a slight visual error is not a big deal, many grips are "ribbed" (i.e., they have an internal ridge/rib running down the back/bottom edge of the grip as an alignment aid for your hands). If you are using ribbed grips, be extra careful to properly align the grip so the rib falls in the right location.
All that's left now is to wipe up any excess grip solvent from the shaft or grip with a paper towel or rag, and place the club upright where the solvent can dry for at least 24 hours before use.
Grip solvent is reusable, so capture the excess and simply return it to the original container. 
You're all done! You've installed your first grip and are ready to repeat as often as necessary. Remember, replace your grips at least every two years if you only play a little, and as much as twice per year if you play a lot. Having proper fitting, proper feeling golf grips will definitely save you strokes, regardless of your game. And why not? Now, you can do it yourself.
