It is the ruler of the Kree. Its name is indicative of the fact that it's the smartest Kree alive, being more intelligent than the greatest Kree scientists and the greatest Kree strategists.
Before making an on-screen appearance in Captain Marvel, Supreme Intelligence was mentioned in Guardians of the Galaxy. The events of Captain Marvel have humbled it to some extent, even if it's not fully reformed, as it's signed a peace treaty with Xandar. Ronan rejecting this peace treaty is what kicks off the events of the movie, and ultimately leads to the formation of the titular Guardians of the Galaxy.
In Captain Marvel, the Supreme Intelligence is a computer that in virtual reality takes a form of someone whoever is using it at the time greatly admires. In the comics and the cartoon, the Supreme Intelligence is a giant Kree head.