What are the different gauges of electrical wire available for homes and how do you determine which gauge you should be using? If it helps, I'm wondering specifically about recommendations as I finish my basement.
This article has a full list and gives examples: http://electrical.about.com/od/wiringcircuitry/a/electwiresizes.htm
For the most part though, 12 gauge wire is used for 20 AMP circuits and 14 gauge wire is used for 15 AMP circuits. You probably can just use the 14 gauge wire in the basement (but it also depends on what you are doing in the basement as far as power requirements). For example, these days a kitchen is always wired with 12 gauge/20 AMP circuits because of all the power demands. The refrigerator will be on its own 20AMP circuit and so will the dishwasher. Then depending on the number of outlets/counter space you may have one of more additional 20 AMP circuits.
The rest of the house is usually 14 gauge but the size of the house will determine the number of 15 AMP circuits for everything.
Also keep in mind that 14 gauge wire is much more flexible than 12 gauge. So don't just go to 12 gauge just because... you will have a much harder time wiring everything if you do.
Really there are only two sizes you'll need to be concerned about.
14 gauge ("white") wire is used for 15amp circuits and is what you will probably use for everything you are doing.
12 gauge ("yellow") wire is used for 20amp circuits. In a typical home these are used in the kitchen and sometimes for dedicated window AC unit circuits. These wires are harder to bend and work with, and also more expensive. Some people use 12 gauge everywhere in order to support space heaters.
There are also "three conductor" versions of both 14 and 12 gauge wire. You probably won't need to use these for anything, but they are most often used for running multiple circuits in a single wire or for connecting 3-way light switches.
There are other sizes as well, but they are generally for specialty use (like for 30/40/60 amp circuits for electric dryers, stoves or hot tubs).
Be sure to check local codes. My county, Geauga County, OH, requires all wiring be rated at 20A, so we cannot use 14 gauge (white romex). All the other posts are correct though that this will be all that you need to worry about unless you are doing something larger in your basement.
A perhaps non-obvious note: make sure the wire you use is rated for at least the maximum load of the associated circuit breaker. That is, if you need to change a circuit from 15 amps to 20 amps, you can't just upgrade the circuit breaker if the wire isn't rated for it.
Unless you're putting together a practice building for firefighters, of course.
Similarly, if you're putting in a circuit that you foresee potentially upgrading in the future, it might make sense to use a beefier wire.
For more detailed information on wiring sizes, you should also check out the National Electrical Code at your local library (in the U.S.) Pretty much every state and municipality has adopted it as the starting point for their electrical requirements. Your locality may have additional requirements as well, though (for example, Chicago requires ALL electrical wire be run in nonflammable conduit). There are a lot of rules regarding box sizes, where to fasten cable, how long wires have to be once inside the box, etc.- all for fire safety reasons.
Off-topic, but all basement circuits except for a sump pump need to have GFCIs installed too.