Full Power

Full power is just a question of connecting the power supply to the braids/ tapes/ track rails - and hence to the motor.  An on-off switch would do this bit! The less resistance at full power the faster the car will go - so low resistance contacts and thick wire help.  For ultimate performance a full power relay may help.

In some circumstances, it is an advantage to have less then full power.  This is provided by a "choke" - usually an adjustable resistor with  a resistance of a small fraction of an ohm.   Link - how to connect a choke   Some controllers have a choke function that's more complicated than a resistor, either reducing the voltage, or sometimes even reducing power for an adjustable length of time after the trigger is depressed.   
(Rather confusingly, "choke" has more than one meaning.   In electronics "choke" is a term for an inductor.   In normal slot racing controller jargon "choke" means a devices for reducing power in which inductance is usually irrelevant.   The more complicated chokes are sometimes referred to as "traction control", in a way that's correct, but they are not doing the same thing as the much more sophisticated traction control systems used in full size vehicles )

Brakes

Brakes are achieved by connecting the motor terminals together. This makes the motor act like a dynamo and slows itself down quickly. The less resistance on brake the sooner the car will stop - so low resistance contacts and thick wire help. This is called "Dynamic braking".

Many drivers find reducing the braking effect can improve the drivability for some cars  at some circuits. The controller can easily provide variable brakes by inserting a variable resistor between the brake band and the negative supply.  Higher resistance gives less braking, disconnecting the  brake wire gives even less.   Link- how to connect a brake box     
With some cars even "no brakes" (the brake wire disconnected) make the car stop too quickly.  Examples of this are group 12 powered cars with extreme gear  ratios and HO cars with a lot of magnet traction.   Less braking  than the brakes disconnected can be achieved by feeding a small amount of power to the motor when the controller is in the brake position.  This is available on some controllers and often referred to as "anti brakes".     The term "coast" is sometimes used for anti brakes, unfortunately "coast" is sometimes used for brake wire disconnected so it is not always clear what people are talking about when they say they have coast.

In the very early days of slot racing controllers didn't have brakes, so the cars just coasted to a halt slowed only by friction.   Some controllers intended for magnet traction cars with heavy magnet traction and some basic home set controllers are made this way.

It is possible to produce more braking by applying a reverse voltage, this was banned from British national rules since the 1960s and is widely banned in other countries.  Is far as I know, there are no commercially produced controllers that have this feature, although I have seen a prototype from the 1960s that never made it into production, 

Cornering

Reducing power for corners is the main area where controllers have developed in recent years.  The normal method of reducing the motor power is to  waste some of the power as heat. All resistance controllers, all diode controllers and most transistorized controllers work this way.

Traditionally a variable resistor has been used to reduce power. This usually consists of many turns of resistance wire wound on a ceramic former, with a wiper making contact direct on the resistance wire. Up till the late 1980s this was the only technology generally available. It worked pretty well, and when the resistor suits the car you are driving the advantages of anything more complicated are marginal .  The major disadvantage is that one resistor didn't suit all cars and all tracks.  Too low a resistance and you cannot go slow enough round some corners - in extreme cases (e.g. Scalextric car on 5 ohms) you end up with something not much better than an on-off switch! Too much resistance and all the action happens too near the full power - in extreme cases (e.g. Group 12 car on 25 ohms) you again end up with something not much better than an on-off switch! Some people had several different controllers - Some people found ways of adjusting the resistance - but the problem was really solved by the introduction of transistorized controllers which were very widely used by the early 1990's.  There is more on resistance controllers here. 

There is an alternative approach to all this hot stuff - that is to switch the power on and off very quickly - if this is done fast enough (say hundreds of times or more a second) the motor behaves just as if the power has been reduced by more conventional means. This method produces very little heat (and wastes very little track power). The wasting very little power is vital for electric powered radio controlled cars that carry their batteries in the car - so "switching" controllers are very widely used in RC cars. Wasting heat is not a problem to slot racers as long as it can be dissipated adequately (simply a question of large enough / high enough power rating resistors, and enough heat sinks on the transistors.) It's half jokingly observed that many club rooms need warming up in winter!