I know there are already quite a few great bottle cutting instructables but I find this technique to be the fastest, and in my expirience the cleanest and easiest way to cut bottles. My project was to make a set of drinking glasses from old pepsi bottles that I had lying around, but you can use this technique for whatever project you like.
Step 1: Tools
All you really need for this is a simple glass cutter you pick up from a hardware store and a propane torch with a relatively fine tip. It may work with something else like a lighter, candle, larger torch etc. but I have yet to try those as my little benzomatic pencil torch has worked so well. My preferred way to do this also involves a lathe with a very slow speed but it can be avoided as I show later in the instructable. All you really need is some way to rotate a bottle at a relatively slow controlled speed, Ive done this on a potters wheel also but you could easily rig something up with a geared motor from a toy car, its only needs low torque.
Step 2: Nick
This is what makes this method so much easier, you dont need some jig to cut around the edge of a bottle perfectly all you need is a to make a little (like 1/8") nick with the glass cutter that would follow around the circumference of the bottle. It should be a decent nick with quite a bit of pressure applied to the cutter.