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 10 years and almost 25000 km, but heavier and faster recumbents finish them after several thousands.
Effectivity of these is somewhere between rims and disks. They are usually somewhat heavier, but weatherproof, almost 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. But these don't run dry - their braking elements are made of bronze and they must be greased to prevent seizing or wearing out quickly.
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.
Recently I got an opportunity to work on hydraulic brakes for the first time: Magura HS11, which are probably the only existing rim hydraulics in the world. They use the same mounting studs as V-brakes. To remove a wheel, instead of unhooking a cable we pull a quick release lever and unhook one actuator. Hoses have the same outer diametre as bowden cables, but no metal reinforcements inside, just homogenous plastic; flexible, but rather hard. To insert a barbed fitting requires roughly the same force as driving a nail, it's impossible by hand, and heating the hose to soften it destroys it. I found an easy procedure somewhere on the internet: clamp the hose in a vise using a wooden block with a hole and slit, insert a piece of an old spoke with a nipple through the fitting and the hose to keep everything coaxial, and hammer the nipple until the fitting comes in:
These barbed fittings are officially not reusable, but if you are careful, the hose can be removed with a knife. Simplified diagram of the hydraulic circuit:
1 - bleeding screw, 2 - filling screw, 3 - adjusting screw
Magura uses mineral oil for hydraulic fluid, their brand is called Royal Blood (probably due to its blue colour). Filling and bleeding is the same procedure, the only difference is the amount of fluid needed. You need two syringes, ideally about 30 ml. Connect a piece of any suitable hose and a barbed fitting to one of them and fill it with fluid so that there are no bubbles inside. Pull the other's plunger as far out as it will go and drill a small hole just below it.
Begin by loosening the adjusting screw 3 all the way out and turning brake lever up so that the bleeding screw 1 becomes the highest point of the system. Filling screw 2 should be at the lowest point (it's OK if the bike stands normally, no need to tilt it). Replace filling screw with the full syringe, then replace bleeding screw with the empty one. Push the fluid in until it starts flowing into the empty syringe (air leaves through the hole you have drilled). Flick the brake lever several times to expel bubbles and continue filling. When the circuit is full of fluid and all bubbles are out, cover the hole in the top syringe, remove it and return bleeding screw to its place. Then remove the bottom syringe and return the filling screw. Leftover fluid can be returned to the bottle for reuse, going through the hoses didn't harm it. Finally adjust brake pad distance by the adjusting screw and you're done.
The only regular maintenance is gradual tightening of the adjusting screws as the pads wear, and loosening it all the way back when they are eventually replaced. The pads are removed by pulling them in the direction of their travel, no tools needed. Their position is set at the actuator level and doesn't need further adjustments. Braking power is similar to well-adjusted V-brakes, movement is as smooth as with a short, brand new and optimally routed cable, and it doesn't become draggy over time. The fluid does not age nor absorb moisture, so it should last theoretically forever. If you heard rumors that hydro-brake bikes must not be laid on their side or turned upside down, they certainly don't apply to these particular brakes: the whole system is fully sealed and can work in any position. Even pulling the brake lever when wheel is out is safe, the pistons can't fall out.