timer0 controls pins 5 and 6 and also is the "clock" used to implement micros(), millis(), delay() and delayMicroseconds(). It runs in "fast PWM mode" which counts only up and cycles every 256 prescaled clocks
The other timers run in "phase correct PWM mode" - they count up then down then up and cycle every 510 prescaled clocks.
Each timer can have its mode changed from the default, but if you change timer0 you'll break micros(), millis(), delay() and delayMicroseconds().
timer1 is 16 bit (but by default is configured as 8 bits). It controls pins 9 and 10. timer2 is 8 bit and controls pins 11 and 3.
For the Arduino Megas there are more 16 bit timers available and the pin assignments are different.