Garter stitch is the pattern that ensues when each row of a knitted fabric is worked over using only the knit (or only the purl) stitch.
What it is
Since the knit stitch creates a "bump" or "ridge" on the opposite side of the work, knitting every row using the knit stitch creates an alternating pattern of ridges and smooth sections. The "bumps" stand out more than the smooth sections, giving the impression that the work consists entirely of wavy ridges.
The finished product contains ridges on both sides (a smooth row on one side is a ridge on the opposite side), making it thicker than many other stitches — and making the stitch ideal for knitted cleaning materials such as dishcloths.
What it's not
The back or "wrong side" of a product knitted entirely in stockinette stitch consists entirely of ridges and therefore resembles garter stitch in some respects. It is not garter stitch.
Why it's handy
Garter stitch is, as the writeups above note, the easiest stitch in knitting because it can be executed as soon as the knitter has mastered the knit stitch. It is frequently used in patterns for beginners as a result.
Unlike stockinette, garter stitch does not curl. It's therefore a popular choice for borders on items that use stockinette.
The exception
If you are knitting in the round (knitting on circular needles with your work joined into a continuous round shape), you must alternately knit and purl rows in order to produce garter stitch.
The name
The stitch was apparently named for its resemblance to the shape of a garter. Which is pretty obvious when you think about it, but we'd just learned about garter snakes in school when my mom taught it to me.
References:
http://www.knittinghelp.com/videos/knit-stitch
http://www.dummies.com/how-to/content/how-to-knit-the-garter-stitch.html
http://knitting.about.com/od/stitchglossary/ig/Stitch-Gallery/Garter-Stitch.htm
BrevityQuest12
278 words not including the reference section and this sentence.