A bicycle brake may work at the rim, or at the hub. Rim brakes have the advantages of light weight, large heat-dissipating area, low stress on the bicycle frame, fork and wheel -- though heat dissipation is limited by risk to the tire.
Hub brakes are more weatherproof, and and are not affected by rim dents or wheel true. Because heat-dissipating area is smaller than with rim brakes, these brakes run hot -- but some are designed to.
Bicycle brakes are operated by hand levers by way of cables, or sometimes hydraulic lines -- except for one type of rear hub brake, called in British English a "backpedaling brake" or "foot brake". In American English, it is called a "coaster brake", an expression which dates back to the its introduction in the late 19th century -- also allowing coasting, unlike a fixed gear.
There is also a page about cables on this site. Poor brake performance often results from cable problems.
What Brakes Would Work Best for Me?
Different types of riding favor different brakes.
Rim brakes are lightest because rims also serve as brake discs. Wheels must be true for rim brakes to work smoothly -- and so, their maintenance also involves wheel maintenance. If rims are wet, rim-brake performance suffers until the brake shoes wipe them dry -- a very serious problem with chromed steel rims, much less a problem with aluminum alloy rims. Rim brakes wear rims, quickly in muddy or sandy conditions. Also see the article about rim brakes-- which links in turn to articles about the several different kinds.
Though drum brakes all are very weather-resistant, their performance varies widely. They are common on utility bicycles, because of their weatherproofness, but only a few special drum brakes can avoid overheating on long downgrades. Generally, the larger the drum, the stronger the braking, and the better heat dissipation. A drum brake which is integrated into a hub can suffer contamination from lubricants. If the drum wears out, the hub, and usually the wheel -- must be replaced. These problems do not occur if the drum brake is external to the hub. (Drum brake article is in the works...)
The Shimano Rollerbrake is a variety of drum brake which is strong for its size. It attaches to the hub with a special fitting which is only found on Shimano Nexus and Nexave hubs. As an external brake, it can be replaced separately from the hub. Several sizes are available. Also see the article about Rollerbrakes.
Disc brakes dissipate heat better than most drum brakes. Disc brakes have become popular on mountain bikes and are gaining popularity for other bicycles. Because only the discis attached to the wheel, a disc brake easily allows wheel changes. Disc brakes can only be installed on frames with special fittings, and have unique problems, worst among which is that a front disc brake can pull the wheel out of the fork. Also see the article about disc brakes.
Coaster brakes are operated by backpedaling, and so can be installed only on the rear wheel. Coaster brakes are the most maintenance-free, and suitable for children with limited hand strength. On a folding or take-apart bicycle, a coaster brake avoids the need for a cable connection. A coaster brake complicates starting and stopping, and prevents the use of toeclips and straps, clipless pedals or derailer gearing. A coaster brake may, however. be incorporated into an internal-gear hub --- but then, except with a two-speed kickback hub, a shifter cable is needed. A coaster brake is OK for utility cycling but unsuitable for speed control on long downhill runs. Also see the article about coaster brakes. A coaster brake is often installed as the only brake on a bicycle -- not a good idea, and that leads to our next topic.
Gimme Two Brakes!
With only a couple of exceptions, every bicycle should be equipped with two brakes -- front, and rear. Any brake will fail sooner or later, and then you really need the other one!
Notable exceptions are:
fixed-gear road bicycles. These can get by with only a front brake because the rear wheel can be slowed by pushing back against the turning pedals;
track racing bicycles, brakeless fixed-gear bicycles, because these are the only traffic on the track, and abrupt stops would be hazardous;
tandems and other bicycles which carry a heavy load. These generally should have three brakes -- powerful front and rear rim brakes for stops, and a drum or disc brake capable of heavy heat dissipation to control speed on descents. See the page on tandem brakes.
Adult tricycles. (OK, not exactly bicycles.) Recumbent "tadpole" tricycles, with two wheels in the front, generally have dual handbrakes -- one on each front wheel, controlled by the hand lever on its side, allowing the brakes to help with cornering. Racing "delta" tricycles, with the two wheels in the back, often have two handbrakes, both operating on the front wheel. Tricycles may have only one rear wheel driven, and only that wheel may have a brake. For safety's sake, there should always be a brake on the front wheel.