If you buy a headset that plugs into the sound port (where the speakers plug in.. little round plug) then you won't need any additional software.
If you buy a headset that plugs into the USB port (small rectangular port) then you'll need the driver software, which will come with the headset.
Several years ago I spent $90 on a set of LTB USB Surround Sound headphones with mic. The sound is absolutely incredible! (Sitting in a GV I can locate the position of an enemy tank just by sound). They still work pretty well after all of this time, but have a little bit of a crackle (which goes away if I bend the wires just right... probably about to break a wire) and one of the headphones is held in place with a rubber band. Also I've never had any problems with the mic.
I know I'll be shopping for a new pair soon, but $90 for 4+ years of service.. well I can't complain.
Now you CAN download teamspeak for other games, but it's not necessary for AH, because (as Gooss said) it had voice communications built into the game.