What's the difference/relationship between Arduino and AVR?
Arduino is a prototiping board, and also the term "Arduino" is used to referr to the IDE and library on the PC side, and all its ecosystem.
AVR is the architecture (developed by atmel) of the microcontroller chip used in all official 8bit board, and almost all clone.
Arduino UNO and 2009, the most used, use the AtMega328P chip.

Many times, Arduino is used to quick test some idea, sensor, and circuitry, then a stand-alone board is builded around the AtMega chip, as it cost 1/10 of the Arduino board, soldered ciurcuit on a stripboard or on a custom PCB are more reitable, and can be optimized on some aspect, like power utlization, space occupied, high current/voltage, and so on.

Newest and advanced arduino board use different chip with very different architecture; the arduino yun use a SAM plus a classic AVR, the due use an ARM (same architecture used by many smartphone), the galileo use an x86 (like a classic pre-multicore cpu)