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, size, and cost 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.

Flash |
RAM |
EEPROM |
1 chip |
10 chips |
25 chips |
|
| ATtiny13A | 1k |
64 |
64 |
$1.40 |
$1.20 |
$1.07 |
| ATtiny25 | 2k |
128 |
128 |
$1.66 |
$1.42 |
$1.15 |
| ATtiny45 | 4k |
256 |
256 |
$2.13 |
$1.73 |
$1.34 |
| ATtiny85 | 8k |
512 |
512 |
$2.84 |
$2.31 |
$1.78 |
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.

Flash |
RAM |
EEPROM |
1 chip |
10 chips |
25 chips |
|
| ATtiny24 | 2k |
128 |
128 |
$1.91 |
$1.55 |
$1.20 |
| ATtiny44 | 4k |
256 |
256 |
$2.15 |
$1.87 |
$1.59 |
| ATtiny84 | 8k |
512 |
512 |
$2.90 |
$2.52 |
$2.15 |
ATtiny2313

Flash |
RAM |
EEPROM |
1 chip |
10 chips |
25 chips |
|
| ATtiny2313 | 2k |
128 |
128 |
$2.26 |
$1.84 |
$1.42 |
| ATtiny4313 | 4k |
256 |
256 |
$?.?? |
$?.?? |
$?.?? |
The Arduino line.

Flash |
RAM |
EEPROM |
1 chip |
10 chips |
25 chips |
|
| ATmega48PA | 4k |
½k |
¼k |
$2.58 |
$2.22 |
$1.86 |
| ATmega88PA | 8k |
1k |
½k |
$3.87 |
$3.15 |
$2.43 |
| ATmega168PA | 16k |
1k |
½k |
$4.08 |
$3.77 |
$3.41 |
| ATmega328P | 32k |
2k |
1k |
$4.30 |
$3.70 |
$3.30 |
ATmega644P

Flash |
RAM |
EEPROM |
1 chip |
10 chips |
25 chips |
|
| ATmega164P | 16k |
1k |
½k |
$4.82 |
$4.14 |
$3.56 |
| ATmega324P | 32k |
2k |
1k |
$6.02 |
$5.18 |
$4.45 |
| ATmega644P | 64k |
4k |
2k |
$7.76 |
$7.22 |
$5.70 |