What are the different gauges of electrical wire available?
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.