An Uchiha will first gain their Sharingan with usually one tomoe when their life is in danger and their Sharingan will continue to get more powerful as they gain more tomoes (maximum of 3 in each eye).

Sharingan abilities

Copying ability
The ability for which the Sharingan is famed. The eyes allow for anyone with them to duplicate almost any technique performed against them as long as they see the hand seals in motion with their Sharingan activated, however it cannot copy summonings or other bloodline techniques (which would require you to have been born with special characteristics). The Sharingan also allows the ability to copy body movements. This is a two-edged sword though, for example: it allowed Sasuke to copy the martial arts of Lee, it did not however give his body the enhanced stamina, so he tired quickly.

Prediction Ability
The Sharingan is much more sensitive to movement than a normal eye, giving the user the ability to detect even the slightest movement in view. With good judgment, this allows the user to predict movement. This is a very deadly ability in hand to hand combat, though the Sharingan user must be able to keep up with the speed of their opponent for it to be effective.

Genjutsu Ability
The Sharingan is able to cast genjutsu through eye contact with an opponent, this genjutsu only requires much skill and no hand seals are required. It is an advanced ability that requires much practice to use effectively. A more adept Sharingan user is also able to instantly realize when they are placed in a genjutsu, although they are still effected by it.

Chakra Sight
The Sharingan is able to see chakra which is naturally invisible to a normal eye like the Byakugan from which it was born, although the Sharingan does not give the insight of the Byakugan. This is also related to the user's ability to predict movement as they can see the flow of chakra through the opponent's body.

Notable Users:
Itachi Uchiha
Kakashi Hatake
Obito Uchiha
Sasuke Uchiha
Madara Uchiha

Mangekyou Sharingan

Madara's Eternal Mangekyou

The Mangekyou Sharingan is an evolved version of the Sharingan, appearing differently for each person whom obtains it. It can be obtained in more than one way, but only one way has been revealed, and that is by killing your best friend. Itachi has displayed three of the Mangekyou Sharingan's abilities, the first is "Tsukiyomi" which allows Itachi to invade one's mind by making eye contact and trap him/her in a world of his design that seems to last 72 hours, but in reality only last 3 seconds. The second he used was "Amaterasu," engulfing whatever he looks at in a black flame which burns anything, even fire. The third is "Susanoo" which takes the form of a giant humanoid that wields the Totsuka sword and the Yata mirror. The sword seals whomever it strikes in a dream-like world, and the shield is capable of deflecting any attack.

Kakashi has attained the Mangekyou Sharingan by currently unknown means and used a mysterious technique to create a miniature black hole wherever he looks which transports matter to another dimension.

The downside to the Mangekyou Sharingan is that it's techniques require a great deal of chakra, and over-use will cause the user to go blind. However, Madara discovered that by obtaining the eyes of another Mangekyou Sharingan user that he'd gain a new Mangekyou Sharingan that'll never lose it's "light" and is able to use a 4th technique. It was later learned that not anyone would do, it has to be from their sibling.

Notable users:
Itachi Uchiha
Kakashi Hatake
Sasuke Uchiha
Madara Uchiha

::NOTES::

Kakashi was not born with his Sharingan eye, but was surgically implanted in him when Obito (supposedly) died.

Sharingan means "Copying Wheel Eye" in Japanese.

Mangekyou means "Kaleidescope" in Japanese.

Itachi's Mangekyou Sharingan techniques are named after Japanese gods that are also siblings: Amaterasu the sun goddess, Tsukiyomi the moon god, and Susanoo the storm god. They were created by Izanagi after he returned from the underworld. Amaterasu and Tsukiyomi were created when he washed his eyes and Susanoo was created when he washed his nose.