In essence, Legilimency is reading and feeling someone else's mind. It is a rare ability to have, and must be worked at for years. It is more than reading someone's mind, but it is that as well. It is knowing another person's mind.
Severus Snape is a very talented Legilimens on top of being a very accomplished Occlumens. By having the skill/power of Legilimency, he is capable of probing into another's mind, which he must do in order to teach Harry how to block such an attack.
Voldemort and Dumbledore are also very skillful in the art of Legilimency. They can use it to tell whether or not someone is lying, for psychological torture (Voldemort's weapon of choice), and for gaining new perspectives (Dumbledore's weapon of choice).