A gear puller is a simple device that allows someone to exert tremendous pulling force on a gear or wheel. This is especially the case when the gear or wheel is mounted flush up against another surface and it difficult to grab the edges or back of the gear. Most the DIYers out there will encounter this while getting their gearhead on--I can't think of an amateur need for one of these outside of automobiles.
There are two main types of gear pullers available: jaw and bolt based. The only difference is the mechanism with which they hold the gear.
Jaw gear pullers usually have three, but sometimes two, arms to hold the edge of the gear. These are normally curved to help reach around to the back surface of the gear, with something of a barb or lip on the end to help grab the edge. The annoying downside is that these arms flail about as you're trying to place the gear puller squarely on the gear, requiring a lot of time do properly. The infuriating downside is the fact that these arms have to octopus outwards around the gear before being tightened in place. This is sometimes a show stopper when working on gears in cramped quarters, especially on those opposed engine ricers.
The second type of puller is my preferred way to have a good pull. It only works on gears that have threaded holes drilled into their surface.Think of a peace sign--the vertical line with those two little diagonal lines coming off the bottom? Make an outline of that shape with a hole in the very center where all the lines intersect. That's the shape of the brace. To make this puller work, you drop two or three bolts into the brace, in a pattern corresponding to the threaded holes on the surface of the gear.
The main reason I prefer this type of puller is the accessibility it offers. There will be cases where, even after you manage to wrap the jaws around a gear, the end of the arms are too thick to hook around the back of the gear. If the gear is mounted too close to another surface, a jaw gear puller is useless. You've been warned.
It's a good idea to get started by screwing in the the center bolt--yes, that huge ass one in the kit. Make sure the end with the hex bolt goes away from the gear--we'll get to it in detail later. Now hold the whole assembly in front of the gear and secure it in place. Either swing the jaws behind the gear or screw those two or three bolts into the face of the gear. Using your fingers, begin to tighten down the large, central bolt. If you are using a jaw setup, tighten the bolt until the whole assembly holds itself in place without your help. Otherwise, just turn the large bolt until the brace starts to press up against the end of the small bolts you've fastened to the face of the gear.
With the puller placed on the gear, make sure that you'll be pulling directly outwards. The brace needs to be perfectly parallel to the face of the gear; adjust the jaws or bolts as needed to properly seat the puller. Then simply put a wrench of your choice on the hex head of that long bolt and start turning to the right. This isn't a race, so proceed gently and constantly check that you are in fact pulling the gear out perpendicular to whatever surface is behind the gear.
Congratulation, you have a gear in your hand!