Bike parts which save our life every day deserve a proper care.
Probably the most widespread type. Their advantage is being light, cheap and usually quite reliable. A disadvantage is they wear rims, they are weaker in snow or mud and they can overheat rims on long descents, up to a point where tyres burst. Their effectiveness range widely from very good (modern V-brakes) to poor (old calipers).
Brake shoes must be in good condition - replace them before their knobs disappear completely. Regularly remove collected mud and dust to prevent grinding the rims too much. But some grinding always remains and rims need to be replaced before their sidewalls become dangerously thin (today's rims have a notch machined in the sidewalls: as long as it's still there, it's good).
The brakes must be adjusted so that the shoes touch rims with their whole surface, don't overlap rim edges (at least not much) and don't rub on the tyre. When the control lever is released, the brake must not drag on anything. When the lever is pulled, the brake must engage soon enough - if the lever hits the handlebar, it's way too late. Everything should be adjustable with a wrench or two. Every brand is slightly different, so it's hard to describe all of them generally - experiment until it starts to work.
Historic note: rubber shoes on steel rims brake very well when dry, but very poorly when wet. Aluminium is more reliable, it brakes more consistently in different conditions.
Probably the strongest type you can find today. They're a bit heavier than any rim brake because you need a rotor in addition to the caliper. But they don't eat rims, can survive higher temperatures and collect less dirt. Brake pads come in two flavours: organic and sintered. The former are cheaper and grind the disks less, but can lose all braking power when overheated. The latter should withstand higher temperatures (better choice for mountains). Principles of operation also come in pair: mechanic (cable-operated) and hydraulic. The former are easier to install or repair by a hobby mechanic, the latter work perfectly smooth regardless of the tube length. I can't compare reliability of the two because I have never tried hydraulics.
Adjusting the pads is a precision work - the travel is small, they are just a millimetre or so from the disk when not engaged. But it's not a problem as long as you have the right combination of bosses, adapters and calipers.
Service life depends on intensity of use: stock pads are still in use on my folder after 8 years and almost 20000 km, but heavier and faster recumbents finish them after several thousands.
Effectivity of these is somewhere between rims and disks. They are usually quite heavy, but weatherproof, maintenance-free and the most durable of all. Their disadvantage is overheating and fading on long descents (the same as disks).
The only element to adjust is the cable stop, so there is not much to write about.
Backpedal brake is a drum brake too, as well as Shimano's roller brake.
These are very old and rare to see (fortunately). Usually they consist of rubber block that gets pressed against the tyre by a lever or cable. Stopping power is poor, even worse in wet and zero in snow.
The only maintenance to do is to replace the rubber block when worn. They can still be found in some bike shops, but I don't know if someone still produces them. But if you want to ride that bike instead of displaying it in a museum, you'd better get rid of this pseudobrake and replace it with something that can really stop you.