PTT stands for Push-to-talk or Press-to-transmit (depending on where you are when you ask). When using a communication system that is half duplex (that is, when one party is sending all others must receive) it serves to manage traffic.
PTT originated with voice radio and survives into the digital communications era. In a voice communications system where more than one user shares a transmit frequency or other medium, the transmitters must by default be turned off to prevent interference. When a user is ready to broadcast, they activate a momentary switch which turns on their transmitter, send their message, and release the switch. In most radio sets, this switch is a button of some type which is, you guessed it, pressed to talk. Typically, if the receiver is on the same frequency as the transmitter, the receiver is disabled while PTT is active.
Some common uses of PTT include aviation radio, CB radio, walkie-talkies and cell phones with PTT features. Those folks yelling at their cell phones on the bus with the loud BEEEEP in between messages are using PTT.
Alternatives to PTT include always-on transmitters, or voice activated transmission.