While many different companies and manufacturers produce mountain bike brakes, there are just a few main types: rim brakes, mechanical disc brakes, and hydraulic disk brakes.
Rim brakes, including Y-brakes and cantilever brakes, are an older style of brake that is no longer found on new mountain bikes today. You may find still find them on cyclocross bikes and road bikes, but even in those applications rim brakes are being replaced by disc brakes However, many older mountain bikes still run these brakes, and perform fine. Almost all rim brakes are mechanical, meaning they are cable-actuated.
The most affordable type of mountain bike disc brake is a mechanical brake. These brakes are cable-actuated, and are much easier to service and maintain than hydraulic disc brakes.
While mechanical brakes are more affordable, hydraulic disc brakes are the most popular, and are generally considered to be the best. Hydraulic brakes, sometimes referred to as "hydros," can provide more stopping power, as well as better modulation than mechanical brakes.
Popular brake manufacturers include Avid, Shimano, Formula, Magura, and Hayes. When researching your next brake purchase, it's also a good idea to examine the best brake cables, brake hoses and fittings, brake levers, brake calipers, brake rotors, and brake pads.