Normally bicycles have mechanical brakes on both the front and rear wheels, although a fixed wheel bike can be legally used on the road with just one brake, usually the front. Again because your legs count as the other brake.
Spoon Brakes
Since the very earliest bicycles appeared at the end of the 19th century, stopping was recognised as important as going. The first brake was a lever that pushed a metal shoe onto the front tyre. The shoe was shaped like a spoon, hence its name of “spoon brake”. It worked fine with solid tyres but became obsolete with the invention of pneumatic tyres.
Rod Brakes
The next type of brake, the Rod brake, was a lever which pulled a rubber shoe against the inner surface of the rim. Some basic and utility bikes had these into the 1980s.
Rim Brakes
The main type of brake for the last 100 years has been the cable-operated rim brake. There are many different designs and some have lasted well for certain uses. The main road brake was the centre-pull brake, followed by various side-pull and then dual-pivot brakes. Tourers, tandems, early mountain bikes and cyclocross bikes used cantilever brakes.
Disc Brakes
Mountain biking ushered in the mainstream use of disc brakes which were quickly taken up by tandems but took much longer to migrate to road bikes. This is because their better stopping power and all-weather performance is at the expense of weight, complexity, cost, and the need for stronger frames. Also the through-axle waves goodbye to the quick wheel-change in a race.
Finally, Dutch and utility bikes often have a form of enclosed drum brake that is either cable operated (just like on a motorbike) or by pedalling backwards. My father-in-law told the funny story of some visiting Dutch students who found out the hard way that British bikes have lever-operated brakes instead of rear-wheel coaster brakes. Apparently, it was quite a sight when the group whizzed down a hill in Bristol, furiously pedalling backwards! Luckily no bicycles or students were hurt.
If you want to discover more about brakes then pop over to the legendary, late Sheldon Brown’s website.