What kinds of brake designs exist, and what are their general pros and cons?
Here's a more complete list, arranged roughly in increasing TYPICAL stopping power, with pictures:

Rod




Largely outdated
Very low stopping power
Imprecise and heavy
For more information: http://www.bikejuju.com/2010/how-do-rod-brakes-work-a-visual-primer/


Coaster/Drum/Hub




Include 'Back-Pedal' Brake
Generally not very powerful
Very low heat produced for extended periods of braking, used in freight bicycles for example
Isolated from the external environment (rain, mud, etc.)
Very low maintenance
Braking force transferred through spokes like disc brakes (Can't use radial spoking)


Side Pull Caliper




One of the weaker brakes out there for Mountain/BMX use
Somewhat common on older cheap bikes
Note the single pivot point
Sometimes used on road
Light weight
Cheap
Simple


Dual-Pivot Side-Pull Caliper




Common on road bikes
Easily centered so they don't drag against the rim
Note the second pivot point on the right arm
Light weight
Aerodynamic
Slightly more powerful than the cheaper calliper brakes


Center-Pull Caliper




Somewhat common on road bikes
Light weight
Protrude less from frame/fork
Pose a cable pull design challenge for suspension designs (both front and rear)


Center-Pull Cantilever




Better braking power than calliper brakes
Require Cantilever/V-Brake mounts on fork/frame (below rim)
Inexpensive
Protrude from frame/fork very extensively, so not used much in BMX / Trials bikes
Pose a cable pull design challenge for suspension designs (both front and rear)
Center better than Caliper brakes


Delta Brakes




Found on some older road bikes, now uncommon
Aerodynamic
While they look like center pull calipers, they have a few mechanical linkages internally that change mechanical advantage and other qualities
Heavy
Variable Mechanical Advantage (not desirable)


Side Pull U-Brakes




Common on some BMX Bikes, older mountain bikes
Common for trick bikes that route the front brake cable through the steerer tube
Require U-Brake mounts on fork/frame (above rim)
Somewhere between caliper and V-Brakes for stopping power*
Very little protrusion from the sides, so very little to catch your leg/pants on


Center/Dual Pull U-Brakes




Common on some BMX Bikes, older mountain bikes
Require U-Brake mounts on fork/frame (above rim)
Even less protrusion than side pulls
Again, between caliper and V-Brakes for stopping power*


Note on U-Brake power: I have a type of center pull U-Brake that uses a wedge that forces the pads into the rim and stops with more force than any V-Brake I've ever used, and has crushed rims for me. The TYPICAL U-Brake design would be between TYPICAL caliper and TYPICAL V-Brakes.



Roller Cam Brakes




Shown here on U-Brake Mounts
Smooth feel
Somewhat rare


Direct-Pull Cantilever Brakes / "V-Brakes"




Most common brake for Mountain bike use
Require Cantilever/V-brake mounts on fork/frame (below rim)
Generally more powerful than non-disc brakes due to the mechanical advantage
Require different levers than the other rim brakes (the levers pull more cable and have less mechanical advantage than the others)


Hydraulic Cantilever Brakes




Relatively New
Increased stopping power over non-hydraulic cantilevers, but lighter and more aerodynamic than disc brakes


Cable Disc Brakes




Don't heat the rim or use it as a braking surface
Will run fine on bent/cracked rims
Aren't as affected by mud/water as rim brakes
Increase load on hub/spokes/rim eyelets because the braking force is transferred through them. Shouldn't use Radial spoking with them.
All this is true of other hub/coaster/drum brakes as well.


Hydraulic Disc Brakes




Same as cable disk, but...
Stronger braking than cable disks.
Better feel / less input force
More expensive


Hydraulic Rim Brakes




Common on rear brake of trials bikes and somewhat in Europe, also used extensively for unicycles.
Most stopping power of any brake type
Provide the most 'lockout' - a stiffened disc brake will still allow flex from the hub and spokes, where this will not
Can easily crush most single walled rims