AVRs and PICs are both micro-controllers. Both are very popular among engineers and hobbyists. The AVR family is made by the company Atmel. I've only ever worked with AVRs. PICs are a microcontroller family made my the company Microchip.
Here's an AtMega168:
**broken link removed**
Isn't it cute?
AVRs have a wonderful development platform called Arduino. They are $30.
I never use dev boards. Just the chip on a breadboard with all it's friends.
Here's the ultimate guide to first getting started with AVRs.
Beginning Embedded Electronics
The parallel port programmer for AVRs is $12 from Sparkfun.