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). More advanced Sharingan users are able to upgrade their Sharingan to Mangekyo Sharingans. The Mangekyo Sharingan is a more powerful version of the Sharingan and it grants its user unique and powerful abilities. The appearance of the Mangekyo Sharingan is also unique to each user, so it doesn't have a general appearance. The Sharingan cannot be taught or copied but it is possible to transfer the eye with Sharingan to another person although the person will be unable to deactivate his sharingan.
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). It however may not be able to copy elements that the user cannot use.
But as the Sharingan allows the user to copy the opponent's handseals, it also allows the ability to copy other body movements as well, but this is a two-edged sword, 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.
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. This gives them the ability to clearly see things they normally wouldn't, and 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 physically keep up with the speed of their opponent for it to be effective.
The Sharingan is able to cast genjutsu through eye contact with an opponent, this genjutsu only requires skill, not handseals. 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.
The Sharingan is able to see Chakra which is naturally invisible to a normal eye. 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.
A secret forbidden technique amongst the Uchiha clan attained by gaining the power of a Senju, the user is able to in essence become a physical illusion for a short time, making them momentarily invincible.
The down sides to using Izanagi is that it consumes a large amount of Chakra, and the eye used to cast Izanagi loses it's light and shuts never to be re-opened.
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.
The Mangekyou Sharingan has 3 abilities, the first is Tsukiyomi which allows the user to attack one's mind by making eye contact and trap him/her in an illusionary world of the caster's design that seems to last days, but in reality only last a few seconds. (It has been noted hby Kakashi that only a Sharingan user of the same blood as the caster can break Tsukiyomi)The second is Amaterasu, projecting a black non-extinguishable flame at whatever the user perceives which burns anything, even fire. It is said the flames will burn for seven day and seven night. The third is Susanoo which takes the form of a giant spectral humanoid that surrounds the user. Itachi's wields the Totsuka Sword and the Yata Mirror: the sword seals whomever it stabs in a dream-like world, and the shield is capable of deflecting any attack that hits it. Sasuke's, however, wields a sword-like arrow and a shield-like bow.
Kakashi has attained the Mangekyou Sharingan by currently unknown means and uses a technique called Kamui which creates a miniature black hole wherever he looks that transports things to another dimension.
Sasuke has created a variant of Amaterasu called Kagutsuchi, allowing him to shape and control Amaterasu's flames.
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; Susanoo also drains the user's life-force. 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.
Note: Kakashi and Danzo were not born with their Sharingan eyes, but were surgically implanted into them.
Sharingan means "Copying Wheel Eye" in Japanese.
Mangekyou means "Kaleidescope" in Japanese.
Izanagi is the god of light. After his younger sibling/wife Izanami, the goddess of darkness, died giving birth to Kagutsuchi the fire god, he traveled to the underworld, but she became the ruler of the underworld and couldn't bring her back. He returned and cleansed himself of the realm's impurities. Amaterasu the sun goddess was born from cleaning his left eye, Tsukiyomi the moon god was born from cleansing his right eye, and finally Susanoo the storm god was born from cleansing his nose.
Kakashi's Mangekyou Sharingan technique, Kamui, translates to "God's Majesty."
The Sharingan is a deviation of the Rinnegan.