Sharingan is a dojutsu, and a kekkai genkai that can be accessed (naturally) only by the Uchiha clan when an Uchiha experiences strong emotions. The design of the Sharingan is having 3 tomoe.
Two basic abilites that the Sharingan offers are -
Dōsatsugan - The techniques being copying jutsu, seeing chakra, and advanced eye movement to detect anything.
Saimingan - Genjutsu, taking complete control over a person's body. (Even tailed beasts can be controlled), and also the user of the Sharingan is able to predict an attacker's attacks by careful observation.
The other two powerful and forbidden jutsus are - Izanagi and Izanami. Izanagi allows the user to warp reality and switch between illusion and reality, allowing the user to escape death is certain cases. Izanami traps the victim in an infinite loop of illusion. Each of these leads to the user going completely blind in one eye, each time he uses either of the techniques (Although Madara retained his vision due to him awakening the Rinnegan).
The Sharingan can be evolved into Mangekyō Sharingan by experiencing the trauma of loss, although the repeated use of the Mangekyō will render the user blind. To overcome this, the user can transplant the Mangekyō Sharingan of a sibling into his, thus awakening the Eternal Mangekyō Sharingan. The design of the Mangekyō is unique and specific to a person, and the Eternal Mangekyō merges the designs of the two Mangekyōs.
In addition to the abilities that the usual Sharingan offers, the Mangekyō also offers a wide range of abilities specific to users, like
- the Susanoo (common technique) - gigantic, humanoid avatar made of the user's chakra which surrounds them and fights on their behalf.
- Amaterasu in the case of Itachi - black flames that cannot be extinguished at all, and it is the highest form of Katon.
- Tsukuyomi, in the case of Itachi - a very powerful genjutsu technique,requires eye contact and traps the target in genjutsu which feels like an eternity for the user but it lasts only a second in the real world.
- Kamui, in the case of Obito and Kakashi - a powerful space time technique, allows the user to teleport anywhere (also to the unique Kamui's dimension) and allows the user to be intangible (the user teleporting himself into the dimension partially) and also the Kamui can synchronise with another space–time ninjutsu, granting the user access to the synchronised technique's dimension (in the case of Kaguya and Obito).
- And in Uchiha Shishui's case, the Kotoamatsukami, an extremely powerful genjutsu technique, that manipulates the victim's thoughts completely without anyone, including the target, having the slightest idea that they were being controlled.
- The Choku tomoe, in the cases of Madara and Sasuke, grants them fluidity in their movements when fighting.
- Kagutsuchi, in the case of Sasuke, applies shape transformation to the flames of Ametarasu.
The Sharingan, Mangekyō, and the Eternal Mangekyō can be transplanted to another non-Uchiha, but would not be very efficient as it would demand a great deal of chakra and it could not be deactivated. (Which is why Kakashi wears an eye patch)
Naruto Wikia - Sharingan
Naruto Wikia - Mangekyō Sharingan
Myself, a huge Naruto fan.