Legilimency is, in the Harry Potter continuum, the art of magically penetrating someone's mind to pick up their emotions, memories, and thoughts. It is not exactly synonymous with "mind reading," as practitioners can, at least in some cases, manipulate the victim's mind as well. The art of shielding one's mind from such invasions is known as Occlumency.
Legilimency is performed by pointing one's wand at the target and speaking the incantation "Legilimens!" The spell is most effective when the caster can make eye contact with the target. Very skilled practitioners can perform Legilimency wordlessly and wandlessly.
A practitioner is known as a Legilimens. Lord Voldemort is known to be one of the most skilled Legilimens of the age. Albus Dumbledore and Severus Snape are also very good at it. Interestingly, all three are known to have striking eyes, though the precise quality varies.