The Sharingan ("Copy Wheel Eye") is the Kekkei Genkai of Konoha's Uchiha Clan, known as a dojutsu; "eye power". It turns the user's eye red and adds a certain number of extra pupils ("tomoe" as they're known), dependent on the maturity/skill of the user. There can be up to three pupils.
It grants a variety of abilities, including but not limited to: the ability to see chakra, heightened perception speeds, the ability to cast illusions via hypnosis, etc. etc. Perhaps one of the more well known abilities, such is its namesake, is the ability to copy movements/techniques of an opponent within the physical/mental limitations of the Sharingan user.
In order to gain a Sharingan, the Uchiha member/descendant must have gone through some significant event in their lives, or a powerful emotional condition, which is why these are also known as "eyes that reflect the heart". Non-Uchiha must transplant them into themselves. Uchiha natives can activate and deactivate their eyes at will, transplant recipients can't.
There is a natural evolution to the Sharingan, the Mangekyo Sharingan ("Kaleidoscope Copy Wheel Eye"), gained by Uchihas going through traumatic events. This is a more powerful version of the Sharingan that gives the user incredible abilities such as Susanoo, Amaterasu, and Tsukiyomi. With great power comes the risk of eventual, permanent blindness if the user abuses the eye too much.
Finally, the pinnacle of the Sharingan is the Eternal Mangekyo Sharingan, gained by closely-related Uchihas transplanting their MS to one another. The EMS is functionally similar to the MS, though the risk of blindness is lost.
... that's all I can think of for now, sensei. Anything I missed?