Sensei's Go Terms defines atari as "the state of a stone or group of stones that has only one liberty."

This means that it (or they) are only one move away from being captured.

In Diagram 1, reading from left to right:

Some people say "under atari": it means the same. Some players - particularly begiinners, and people playing with beginners - will announce "Atari" like some players will announce "Check" in Chess. It's a courtesy, but don't expect it. It won't happen in higher levels of play.