Muggles' Guide to Harry Potter/Magic/Legilimency

Overview
Legilimency, a compound of the Latin 'legere' (to read) + 'mens' (mind), is the art of, as one might suppose, reading another person's mind.

Extended Description
Characters like Lord Voldemort and Albus Dumbledore use Legilimency, especially to find out if someone is lying to them. Legilimency does not work well on wizards or witches who are experts in its counterpart, Occlumency (Severus Snape is a good example of this).

Professor Dumbledore orders Harry to take Occlumency lessons from Professor Snape in order to block the communications Harry is receiving mentally from Voldemort; Snape, of course, must then use Legilimency to give Harry something that he can attempt to block. Apart from the Occlumency lessons, Snape will attempt to use Legilimency directly on Draco at Christmas in Harry's sixth year, only to be foiled by Draco's skill at Occlumency; and at the end of Harry Potter and the Half-Blood Prince, where he detects Harry's intent to form spells and counters them. There are several places where Snape apparently uses Legilimency on Harry, without Harry being aware of the possibility; several times in the first three books, starting just before the match against Hufflepuff, Harry has the uneasy feeling that Snape could read his mind. We also believe that Voldemort uses Legilimency to confirm Snape's story about the date of Harry's departure from Privet Drive in Harry Potter and the Deathly Hallows, and when Harry returns to Hogwarts it seems that Snape is trying to use Legilimency on Professor McGonagall to see if Harry is nearby.

Analysis
Legilimency is introduced in Harry Potter and the Order of the Phoenix as being the reason that Occlumency is required. We are introduced to the idea that via the practice of Legilimency, Voldemort can read, if not minds, at least memories. Snape, in the abortive Occlumency classes, draws a distinction between Legilimency and reading minds, but Harry is unable to see what the distinction actually is, and Snape, due to his dislike of Harry, clearly does not want to spend any additional time on attempted explanations. It is possible that the distinction Snape is drawing is that between thoughts, which are extremely fluid, and the results of those thoughts, namely actions, words, senses, and memories. It is certainly true that Snape, in the Occlumency lessons, is able to retrieve memories from Harry's mind, and perceive what he is doing well enough that he is able to block every spell Harry tries to hit him with at the end of Harry Potter and the Half-Blood Prince.

As we find in Harry Potter and the Order of the Phoenix, Harry and Snape hate each other actively enough that Harry is unable to calm his mind, and Snape does not help matters by constantly needling him. As a calm mind is necessary for Occlumency, it would seem that this effort was doomed to fail from the outset. Dumbledore later says that he had made an old man's mistake, assuming that the feud between James and Snape was far enough in the past that Snape could put it aside.

Greater Picture
As mentioned in the entry for Occlumency, it is one of Dumbledore's few mistakes to believe that Occlumency will, in fact, prevent Harry's recurring dreams. Legilimency, the ability to read another's mind, does not include the ability to plant visions in the head of another person, as far as we are told. Additionally, the channel used to transfer those visions, we later find, is a fragment of Voldemort's soul that is attached to Harry. Occlumency may or may not prevent the placement of these visions in Harry's dreams, but it seems unlikely given the information Dumbledore had at the time. However, even to Dumbledore, this was a new area of magic, so we must assume that he was doing the best he could with the tools he had.

One other question that arises is how Snape was able to send his Patronus to Harry in the Forest of Dean, a patch of mixed forest and fields covering some few acres, when Harry's exact position was masked by a large number of protective charms. It is possible that Snape used Legilimency to see through Harry's eyes, then quartered the Forest with his Patronus until Harry saw it.