The password is a four digit code given on the screen.
The screen then functions as a second monitor for the computer. With the Mavericks upgrade installed it is the user's choice whether to mirror the displays or use the projector as a separate screen. Older systems can only mirror displays.
Troubleshooting Apple TV connections:
Make sure the projector is on.
Check the power, ethernet, and HDMI cables for the Apple TV to be sure it is properly connected.
If the first two do not fix the problem, unplug the power to the Apple TV and plug it back in to reboot the device. (Rebooting at the start of the day is not required, but may help if problems occur often.)