This page will not explain why you should use AVR chips or why PIC is better/worse. It is more the other way around - if you want to use an AVR chip - which one is a good choice. Capabilities and size are basically the main factors.

With 8 pins these are the smallest chips. Advertised with 6 digital I/O lines, but I strongly recommend not to use the reset line - which means 5 usable I/O lines. Older chips (like ATtiny11, ATtiny12, ATtiny13, ATtiny15, ATtiny26) should not be used. I do not use the ATtiny13A either, because the fusebits are different and the memory is pretty small (I rather spend the additional 20cents to get a ATtiny25.

ATtinyX5

Flash
RAM
EEPROM
ATtiny13A
1k
64
64
ATtiny25
2k
128
128
ATtiny45
4k
256
256
ATtiny85
8k
512
512

With 14 pins these are very powerful chips, best for most projects. More IO pins than the ATtinyX5 and just a few less than the ATtiny2313.

ATtinyX4

Flash
RAM
EEPROM
ATtiny24
2k
128
128
ATtiny44
4k
256
256
ATtiny84
8k
512
512

ATtiny2313

ATtiny2313

Flash
RAM
EEPROM
ATtiny2313
2k
128
128
ATtiny4313
4k
256
256

The Arduino line.

ATmegaX8

Flash
RAM
EEPROM
ATmega48PA
4k
½k
¼k
ATmega88PA
8k
1k
½k
ATmega168PA
16k
1k
½k
ATmega328P
32k
2k
1k

ATmega644P

ATmega644P

Flash
RAM
EEPROM
ATmega164P
16k
1k
½k
ATmega324P
32k
2k
1k
ATmega644P
64k
4k
2k