A cooling system of some type is required for a computer. Computers without cooling can reach temperatures high enough to cause permanent damage to the computer's components. Motherboards generally have safeguards that turn off the computer when the internal temperature reaches a certain threshold to try to prevent damage, but this is only in the event that the installed cooling fails. Bottom line, without cooling, a computer is inoperable.
There are 3 main types of computer cooling: fans, heatsinks, and water cooling. The most common type of cooling is a hybrid between heatsinks and fans. Computers that are overclocked greatly will require water cooling in order to sustain a safe temperature.
Fans and heatsinks, as noted above, are the most common type of cooling for the simple fact that they are effective and inexpensive. The purpose of fans is to cause air movement within the case, keep the components cool, and, when the case has a cooling system set up to include exhaust abilities, will push the warm air out of the case. Heatsinks act as a dissipator, dissipating heat away from essential components. Generally, there is a heatsink/fan unit mounted on the processor, as well as 1 or 2 case fans. There are also often heatsinks on the memory modules, or RAMsinks, or heatsinks on other important devices on the motherboard.
Water cooling is much more expensive but has the ability to provide much more cooling power than conventional fan and heatsink setups. Another reason to install a watercooling setup is to reduce the sound the computer emits since fans can become distractingly noisy. Water cooling systems are fairly complicated, the basic idea behind them is that there is a source of water. A pump sucks some of the water in and pumps it out of the source through a hose. The hose runs to the radiator, flows through the radiator, and fans low air through the radiator in an attempt to control the temperature of the water and adjust it to the desired temperature.