A microcontroller is essentially a computer in a single IC. It contains usually, at least a CPU, RAM, ROM, and usually a few other features for input/output like perhaps analogue to digital converters etc. all in one IC package. A microprocessor is usually just essentially the CPU itself (maybe a few other features)
Mircocontrollers are extremely useful for embeded systems (computers as part of some larger system, like a robot for example), because they are simple and cheap. Often you dont need any other parts like seperate memory or logic, it all comes built in the microcontroller.
What are they used for?
Many people dont realise, but most computers in the world arent things that sit on your desk with a mouse and keyboard. They are in TVs, DVD players, phones, microwave ovens, cars, traffic lights, industrial machines, telephone networks... basically any moderately complex modern electrical device will probably have a microprocessor or controller in it. I have a simple electric gate motor controller here with a microcontroller in it.