Upon seeing Obito approach him, Mangetsu nods once as a sign of greeting his teacher. Obito then proceeds to warn him about what was to come if one seeks to acquire and learn the sharingan. It was clear in Mangetsu's mind that he had to stay, it would be a disgrace to not even attempt acquiring his own clan's powers. Not to mention, he had plans he wanted to achieve and for that, more power was needed. Once Obito saw that the Sharingan-virgin did not leave his spot after the warning had been given, Obito proceeded to ask him about the Sharingan and its abilities which Mangetsu then replied to.
From my understanding, the sharingan is a dojutsu that can be awakened only amongst members of the Uchiha clan. It has a long story and its original abilities stems from Kaguya Otsutsuki. The sharingan has multiple stages, the greatest being Eternal Mangekyou Sharingan and the first one being the sharingan with a single tomoe which is awakened by experiencing emotional stress. I assume the sharingan wielder experiences emotional stress in one way or another when progressing from stage to stage, such as the first tomoe that is awakened when being in a difficult situation in battle or Mangekyou Sharingan when witnessing the loss of someone close to you.
As for abilities, it allows us to perceive chakra signatures with our eyes, which allows us to distinguish one person from another and the jutsus they perform to a certain extent. It also allows us enhanced perception in the sense that it helps us keep track of faster movements which in turn helps us analyse our opponent's body movement more thoroughly so we can more easily predict how they will move. The sharingan also focuses on illusionary arts, both offensively and defensively. Its wielder is capable of piercing through most illusionary techniques that affect the sense of sight and can also send illusions that were cast on them by an enemy back at them, making them fall to their own trap. These abilities' capabilities vary from what stage the sharingan has. Sharingan also has its own set of illusions, the most unique amongst them being of hypnosis type. Meaning they can force the victim to do certain things in reality rather than simply playing with their 5 senses. Another handy ability would be that of copying other shinobi's movements and techniques by memorizing the hand seals sequence and how the chakra flows within themselves when the technique is being performed.