From BR Bullpen

A baserunner is safe when the defense has failed to put him out, either because he has arrived at a base ahead of a force play, or escaped an attempt to tag him out. On close plays, the nearest umpire will signal this by crossing his hands in front of him and then opening his arms with emphasis while yelling "safe". When there is no play on the runner, the umpire will not even bother to signal as it is obvious that the runner is safe.

A runner who is safe can stay at the base he has just reached if the third out of the inning has not been recorded in the meantime. A special case is when a runner is "safe at home"; in this case his team is credited with a run.