Fin Terminology

Base
The base refers to the length of the area where the fin meets and is secured to the surfboard. The base is important to the drive, lift and acceleration of the surfboard.
Depth
Depth describes how far the fins plunge into the ocean water. Obviously, shorter fins do not grip the water as effectively as larger fins and may result in slippage in some surf. Long fins can feel slower but produce better drive. As a rule of thumb, fins of 5 inches or greater are considered large. Small fins measure at approximately 4 inches or less.
Area
Base and Depth can be used to calculate the area of the fin. You should determine, usually through trial and error, the correct fin size (area) large enough to hold the board on the wave without sliding off  but small enough to allow for effective drive in controlled turns.
Sweep
The sweep of the fin describes how far a fin angles backwards. The further backward the sweep, your turns will be wider or more rounded. A straighter fin enables tighter turns (FCS nomenclature examples are: upright fin = TC Redline; a highly swept back fin = YU).
Cant
The angle for how far your fin leans to one side.
Toe
The toe indicates the angle of the fin. If it is parallel to the board's centerline there is a 0 degree toe angle to the surfboard. The shaper of your surfboard has permanently set this angle.
Foil
Fins, much like airplane wings, are curved which creates a foil effect. Most often the middle or back fins are foiled on both sides. Side fins are commonly flat on the inside and foiled on the outside. Recently a handful of fin designers have used a slight foil on the outside fins (FCS has begun offering fins with inside foil). Hydrodynamically a fin foil causes areas of high and low pressure to move through the wave to keep you in by moving from high pressure to low pressure. This is similar to how a plane stays in the air.
Flex
Stiffer fins drive harder. Softer fins can offer less performance.
Drive Versus Speed
Drive relates to acceleration and maintenance of speed though turns. Speed is how fast you go without which effects the relative ability to turn or stay on a wave.
Stiff Versus Loose
Basically, an overly loose ride happens when your fins are larger than you need. Too stiff a ride occurs when fins are too small. Too loose will slow you down. Too stiff will cause slippage but allow for higher speeds.