Sharingan:

The Sharingan (写輪眼; Literally meaning "Copy Wheel Eye", Meaning (Viz) "Mirror Wheel Eye") is a Dōjutsu Kekkei Genkai, which appears in the members of the Uchiha clan. It is one of the three great dōjutsu, along with the Byakugan and the Rinnegan.

Stages:

The Sharingan is not automatically activated in the user by birth. It is only first unlocked under some stressful or emotional condition after which the Sharingan may be used freely. When the Sharingan is first activated by a user, it has a different number of tomoe (巴) around the central pupil, varying between users.^[1]^[2] The wielder of the Sharingan, through continued use and training, evolves it further which involves gaining more tomoe, up to a maximum of three per eye. As the Sharingan evolves, the user gains heightened abilities.

Abilities:

The first of the Sharingan's powers is being able to see the flow of chakra. The Sharingan can tell if a person is under a genjutsu because the person's chakra flow would be irregular.
The Sharingan's second-most prominent ability grants the user an incredible clarity of perception, allowing them to easily recognize genjutsu and different forms of chakra, though not to the same extent as the Byakugan. This also allows the user to pick up on subtle details, enabling them to read lip movements^[3] or mimic something like pencil movements.^[4] As the Sharingan evolves, gaining more tomoe seals, this ability extends to being able to track fast-moving objects before finally giving some amount of predictive capabilities to the user. However, even though the user may be able to see an attack or know it is coming, their body may not have time to react.
The Sharingan's third and most well-known ability is that it grants the user the ability to memorize any technique that he or she witnesses, apart from other Kekkei Genkai techniques. The user can memorize ninjutsu, genjutsu, and taijutsu with near perfect accuracy, allowing the user to use the techniques as his own, or even modify them to create his own new techniques, much like Sasuke Uchiha's Lion Combo. In order to reproduce a copied jutsu, however, one must have the necessary skill or ability to perform them. A prime example is Rock Lee's brand of taijutsu. While Sasuke was able to copy some of Lee's moves, they put a much more significant strain on him since he hasn't trained nearly as much as Lee has. In addition to physical skill and kekkei genkai techniques, the Sharingan cannot reproduce jutsu dependent on other factors, such as summons that the user has not signed a contract with.
The Sharingan's final commonly-used ability is capable of inducing a unique brand of hypnosis that involves suggesting actions and thoughts to the opponent through simple eye contact. When used against animals, the eyes of the target take on the pattern of the users Sharingan while under its influence. When used in conjunction with the other aspects of the Sharingan, this allows the user to perfectly mimic every move their opponents make even before they do, making it appear as if the user can "see the future". More advanced Sharingan users can take the hypnosis ability even further by tricking or even trapping their opponent within an illusion. It also allows for other forms of hypnotism, such as putting an opponent to sleep, replaying memories to someone else, or creating various illusions to fool an opponent. The potency of the Sharingan's illusion is dependent on the user's knowledge and skill with implementing genjutsu.
In addition to these commonly used techniques, the Sharingan allows the user to perform the genjutsu Izanagi. Using Izanagi, the user can warp reality for a short amount of time, changing reality into illusion and illusion into reality. However, this technique has been dubbed kinjutsu, because use of it causes the user to go blind.
Although the Sharingan consumes the user's chakra while active, the chakra levels that the user must exert is so insignificant that they can keep it active constantly. In fact, Itachi Uchiha was so masterful of it, he was able to keep it active for most of the time, only once deactivating it after using Tsukuyomi on Sasuke and Amaterasu to escape Jiraiya. However, when a Sharingan is transplanted to a non-Uchiha, the chakra drain is significant, leading them to cover it up when not in use, to stop the drain.

Receiving the Sharingan:

As already mentioned, the Sharingan needs to be activated by the user himself or herself under a difficult, emotional stress. After its first appearance, the user can use the Sharingan any time he or she pleases. It is also possible to transplant Sharingan from one person to another, as occurred with Kakashi Hatake, who received his left eye's Sharingan from his dying comrade, Obito Uchiha. Because of this, Kakashi's Sharingan cannot be deactivated, and as using the Sharingan requires much more chakra from him than it does from an Uchiha clan member, he usually covers it with his headband to conserve chakra. Also, Kakashi needs to rest for a longer period of time after using his Sharingan extensively in battle. In a similar way, Danzō Shimura had a Sharingan instead of his normal right eye, that he gained from Shisui Uchiha, which he kept hidden under his bandages. Danzō also possessed ten Sharingan in his right arm.

Mangekyō Sharingan:

The Mangekyō Sharingan (万華鏡写輪眼, Mangekyō Sharingan; Literally meaning "Kaleidoscope Copy Wheel Eye") is a heightened form of Sharingan. The Mangekyō Sharingan gives the user access to powerful and forbidden techniques. The Mangekyō Sharingan is distinguished from a normal Sharingan by its appearance, which changes the form of the tomoe seal. Unlike in the case of the Sharingan, the appearance of the Mangekyō Sharingan differs from user to user; Kakashi Hatake's and Itachi Uchiha's both resemble pinwheels, Madara Uchiha's and Izuna's look more like a detailed form of the basic three-tomoe Sharingan, and Sasuke Uchiha's looks like three intersecting ellipses.
The Mangekyō Sharingan is acquired through the death of the person closest to the Sharingan user. To expedite this occurrence, Uchiha throughout history have killed their closest friends in order to meet this criteria,^[1] though how Kakashi gained his Mangekyō Sharingan has yet to be explained. To obtain the Mangekyō Sharingan, one must feel the emotion of losing a friend or family, as merely killing them or watching them die will not activate it.
Over time, use of the Mangekyō Sharingan's techniques deteriorates the user's eyesight to the point of blindness. Only by taking and transplanting the eyes of a sibling can the vision be restored, a process that results in the creation of the Eternal Mangekyō Sharingan (永遠の万華鏡写輪眼, Eien no Mangekyō Sharingan). Not only does this process restore one's eyesight and ensure that the light will never leave the eyes again, but it also gives the person greater power.

Mangekyō Sharingan Variations:

Itachi Uchiha

Itachi Uchiha gained the Mangekyō Sharingan by killing his best friend, Shisui. With the Mangekyō Sharingan, Itachi was able to use at least three powerful jutsu. With his "left Mangekyō" he could use Tsukuyomi, an extremely powerful genjutsu that allowed him to torture opponents for what seemed like days in a matter of seconds. With his "right Mangekyō" he could use Amaterasu, a ninjutsu which creates virtually-inextinguishable black flames at the user's focal point that continue to burn until the target is reduced to ash. The last of these techniques was Susanoo, which allowed Itachi to summon a massive ethereal warrior. This entity fights on behalf of Itachi and shields him from all attacks, rendering Itachi essentially invincible.
Use of the Mangekyō Sharingan put a great strain on Itachi's body and would quickly deplete his chakra reserves. He told Sasuke that he wanted to take his eyes in order to escape these side-effects. After Itachi's death, Sasuke takes his eyes to escape the same side-effects and obtain an Eternal Mangekyō Sharingan.

Kakashi Hatake

Kakashi Hatake developed his own Mangekyō Sharingan during the time-skip. With his Mangekyō Sharingan, Kakashi can use a space-time ninjutsu called Kamui to send a target to another dimension. The attack requires considerable effort to aim precisely and leaves Kakashi's chakra significantly drained; he collapses after using the technique three times in a single day.

Madara Uchiha

Madara Uchiha was the first Uchiha to awaken the Mangekyō Sharingan. He eventually grew blind from overusing it, and only by taking his younger brother, Izuna's, eyes did he regain his sight. This also gave him an "eternal" Mangekyō Sharingan that gave him great power. His Eternal Mangekyō Sharingan looks identical in appearance to Madara's first Mangekyō combined with Izuna's Mangekyō.
Madara, posing as Tobi, has been shown using an unnamed space/time ninjutsu which makes him nearly invulnerable. He can also teleport himself or others to another location. It is unclear if either ability is because of his Mangekyō Sharingan. Awakening the Mangekyō Sharingan also granted him the ability to control the Nine-Tailed Demon Fox. While it is unknown what other (if any) Mangekyō jutsu he may have, Madara has demonstrated detailed knowledge and understanding for each Mangekyō jutsu.

Izuna Uchiha

Izuna Uchiha acquired his Mangekyō Sharingan soon after Madara did. Madara later took Izuna's eyes to regain his lost vision. According to Madara, Izuna willingly gave up his eyes for the clan's sake.

Sasuke Uchiha

Sasuke Uchiha awakened his Mangekyō Sharingan after learning the truth about his brother, Itachi. In his grief over the knowledge that Itachi was not the villain he imagined him to be, coupled with Sasuke's role in Itachi's death, he awakens his Mangekyō Sharingan. Unlike other Sharingan, which have a black design and red background, Sasuke's Mangekyō is the first one that is shown to have a red design and a black background.
Sasuke has been shown using the same Mangekyō Sharingan abilities as Itachi. With his "right Mangekyō" he can use Tsukuyomi to break a target's will and paralyze them.^[2] Danzō Shimura stated that Sasuke's version is not capable of altering the victim's perception of time, and thus is vastly inferior.^[3] With his "left Mangekyō" he can use Amaterasu. It has been implied that his skill with Amaterasu surpasses Itachi's, as he is able to apply Shape Transformation to the flames, and even extinguish them. Sasuke has displayed the ability to use Susanoo, which he can even partially manifest to fit the situation, such as using a ribcage for defense or manifesting only an arm to interact with his surroundings. Sasuke's Susanoo develops in stages during various battles; he first starts using an incomplete version against the Fourth Raikage, develops a version capable of firing arrows while fighting with Danzō, and later forms a version almost identical to Itachi's during his battle with Kakashi.
Like Itachi, using the Mangekyō Sharingan puts a great strain on Sasuke's body and eyes. When he eventually starts to go blind, he has Itachi's eyes transplanted to restore his vision. His "Eternal" Mangekyō has yet to be seen, but he claims he can feel Itachi's power flowing through him.