Most combustion engines, be they gasoline or diesel, use a timing belt or chain. This wraps around the front of the engine and "times" the turn of the crankshaft with the camshaft(s) and other components. In most engines, the timing belt/chain also runs the water pump and sometimes the fuel and oil pumps as well.
Until recently, most engines used a timing chain. In the past two or three decades, however, most automotive manufacturers have been moving to a timing belt instead. There are several reasons for this, but both chains and belts have advantages and disadvantages.
Timing Belt Advantages
The main advantage to the timing belt is that it's much lighter and cheaper than a chain. Timing belts are thinner in both profile and width, weigh far less than a chain, and require fewer parts on the engine to keep them lubricated and tensioned.
Timing belts are also quieter and can often be more precise as well, since they have tighter tolerances for stretching with heat and time.
Timing Chain Advantages
The primary advantage of the chain is its robustness. A timing chain typically lasts the lifetime of the engine and requires no changing. They can also withstand much more pressure and damage before breaking, so when they fail, they tend to fail slowly and cause the engine to die rather than damage itself.
Timing Belt Disadvantages
The primary disadvantage for timing belts is that they require replacement at much shorter intervals than do chains. Replacing a timing belt is expensive and labor intensive. It also requires knowledge and special tools, making it a difficult task for the home mechanic.
Timing Chain Disadvantages
The primary disadvantage of the chain is its weight and noise. Chains are much larger and heavier than belts and a such, they tend to make noise under the hood. They require tensioners and dampers to stay in place and as quiet as possible and require more effort (energy) to turn than does a belt, so they make the engine slightly less efficient.