Foundation piecing -- generally speaking -- is sewing your fabric to something else (foundation) to create your block. The foundation may ... or may not... be left a part of your quilt.
In the case, for example, of crazy quilting, many people use a piece of muslin or something similar to sew their crazy patches to. This is for several reasons. In crazy quilt patches, people often do not limit themselves strictly to quilting cottons. They may be using velvets, satins, sateens, knits, laces... and to keep your fabrics stable... you need something "under them". This actually makes the process of crazy quilting easier!
For other projects, like a mariner's compass (or any of Carol Doak's wonderful stars/blocks)... you deal with some really strange triangular shapes... with very long pointy points. While you can create templates to do this... you will likely be bald by the time you finish (having pulled all your hair out in frustration!) For these kinds of blocks, many people paper piece (a variation of foundation piecing). By sewing to the paper you accomplish several things: 1) you stabilize the very long exposed biases created with these irregular shapes and 2) you can create VERY precise points without going crazy (or bald!) Paper piecing is the basis of other products like Thangles or Triangle papers and a wide variety of patterns where you deal with either very small pieces of fabric, very irregular shaped pieces or a combination of both. (Google for MH Designs to see some of the fun things that you can tackle with relative ease using the paper piecing technique that you might never try with traditional piecing.
Some people swear by paper piecing... some swear at it! I'm in the first group -- I love the way you can get such precision... without going crazy. However, I have picked up a few hints along the way that help. If you wish to try it -- be patient... with paper piecing you have to kind of "think backwards" - so for some quilters it is a little confusing. But once you get your head wrapped around it.... YOU'LL LOVE IT!
I have a few paper piecing projects in my webshots gallery -- probably my favorite is the kimono blocks in my Kimono Quilt...
If you have more questions about Foundation Piecing.... post back... people here will be happy to offer suggestions and helps!