Cortex A (with rather high numbers) are relatively high performance chips, commonly found in one board computers, something like the BeagleBoard.
Cortex M are microcontrollers for embedded usage, slower clocks, but geared towards periphery like CAN, SPI, I2C etc. You can find a lot of those in industrial use.
Cortex R are less common, I think they are geared towards automotive, for real-time stuff, although -M is also used for that.
Above all those sit those big ARM9/ARM11 chips.
Within the M family there are 0 for relatively tiny footprint, 1 is special (synthesized into your FPGA), 3 is the "normal" one, and 4 has DSP capabilities and a bit more performance.