An AVR microcontroller is a type of device manufactured by Atmel, which has particular benefits over other common chips, but first what is a microcontroller?
The easiest way of thinking about it is to compare a microcontroller with your PC, which has a motherboard in it. On that motherboard is a microprocessor (Intel, AMD chips) that provides the intelligence, RAM and EEPROM memories and interfaces to rest of system, like serial ports (mostly USB ports now), disk drives and display interfaces.
A microcontroller has all or most of these features built-in to a single chip, so it doesn’t need a motherboard and many components, LEDs for example, can be connected directly to the AVR. If you tried this with a microprocessor, bang!