Here's a study guide I wrote for beginning figures.
What is a school figure?
A figure is a set pattern of edges and turns based on full circles, and using every type of turn except, I believe, the mohawk. There are around 43 standard patterns and many variations. Figures are skated on 2 or 3 contiguous circles.
I’ve heard it called “Patch.” Why is that?
Because you skate it on a single patch of ice, approximately 6-9 times your height in length, by two to three times your height in width. (Beginners use a smaller circle.)
Who does figures?
Up to 20 years ago, all competitive skaters were required to pass up to 8 Figures tests (all 43 patterns mentioned above). Your free style level was determined not by which jumps you had mastered, but by which Figures tests you had passed. Most coaches born before 1985 or so have had at least some Figures background. US Figure Skating dropped the requirement in 1990. Now only a handful of rinks still teach this skill. If you watch Sarah Hughes' Gold Medal Olympic exhibition program, you'll see her start the program with a figure-- specifically, loops.
So why do it?
Figures help you with proper pushing, body alignment (posture and balance), edge control and clean, controlled turns. This will help you with your jumps, spins and footwork. A properly skated figure is two perfect circles, each one skated 3 times, one exactly on top of the others. Someone really good at it can trace it perfectly! It’s harder than it looks!