Arduino is a prototyping board, and the term "Arduino" is also used to refer to the IDE and library on the PC side, and the whole ecosystem.
AVR is the architecture (developed by Atmel) of the microcontroller chip used in all official 8-bit boards, and almost all clones.
Arduino UNO and 2009, the most used boards, use the AtMega328P chip.
Many times, Arduino is used to quickly test some idea, sensor, and circuitry, then a stand-alone board is built around the AtMega chip, as it cost 1/10 of the Arduino board, soldered circuit on a stripboard or on a custom PCB are more reitable, and can be optimized on some aspect, like power usage, space occupied, high current/voltage, and so on.
Newest and advanced Arduino board use a different chip with a very different architecture; the Arduino Yun uses a SAM plus a classic AVR, the Due uses an ARM (same architecture used by many smartphones), the Galileo use an x86 (like a classic single-core cpu).