You have two sorts, the kind that uses little cleats that sit in a recess in the soles of your shoes, and the kind that uses great big cleats that make you walk like a duck.
SPD: Shimano makes probably the most common system. Fair amount of free movement ("float") to keep knees happy. Easy to adjust the force needed to unclip. Super reliable - my first pair is 15 years old now and still ticking (mostly). Little cleats. Has spawned lots of imitators, some of which are compatible, some of which aren't (probably to get around patent law).
Crank Bros. Egg Beaters: Super minimal mechanism is good for mud and snow. Four-sided design means it's practically infallible for clipping in. Lots of free movement. Little cleats.
Time ATAC: A bit like two-sided Egg Beaters with a bigger platform for your shoe. Probably the second most popular system and has lots of free movement (side to side and twisting). Uses little cleats.
Speedplay Frogs and Zeroes: Tiny little pedals with the cleat housing the mechanism. I have no idea what they're like to ride. Uses big cleats, technically.
Look Keo: Big triangular cleats with pedals that work a bit like ski bindings. Big platform is great for putting power into the pedals but the cleat needs a roadie shoe. Better versions have adjustable clip tension, adjustable axle length, adjustable cleat tilt, adjustable float. Also comes in original Look Delta format with flatter cleats. Exustar makes some pedals that are compatible with Look.
Shimano SPD-SL: Basically the same as Look Keo, but not directly compatible.
Time: Metal hook-shaped cleat in a sort of Look-alike pedal. Generally recommended for anyone with dodgy knees, or for anyone keen to avoid dodgy knees. Also needs a specific shoe last time I looked (no pun intended).