Operational amplifiers (op-amps) are everywhere, in everything, and can do just about anything. Well, okay, maybe not quite, but there is no doubt that they are one of the most useful ICs ever to hit the market.
Developed by Bell Labs, the first op-amp was made with vacuum tubes and released in 1941. Later, in 1968, the uA741 was released and has since become one of the most ubiquitous ICs on the market today. Many companies produce versions of the 741 design, and many more op-amp designs are available. A quick perusal of the datasheets will show that you can have high speed (OP37), multiple monolithic packages (OP484/OP482), or high current output (TCA0372). All of these chips have their strengths and weaknesses, and there are many more to choose from.
But how do you use them? What are they for? For a basic overview of how op-amps can be implemented into your project, check out the Instructables Op-amp Basics Part 1, which covers the simpler concepts like filters and amplifiers, and Part 2, which goes into more advanced uses like integrators and wave form generators.