The Great Old Ones, also known as the Old Ones, the Elder Gods, and the Death Lords, are beings who came from a universe before this one (the works of H.P. Lovecraft and other out-of-copyright authors) to be shoehorned into Doctor Who whenever the writer is out of ideas but wants something really impressive for the Doctor to face.
The Gods of Ragnarok were Great Old Ones. So was the Fendahl, according to the novelisation. The Virgin New Adventures novels later retconned a bunch of other monsters like the Great Intelligence, the Nestene, the Celestial Toymaker, and Adric into being Great Old Ones, and directly borrowed beings like Cthulhu and Azathoth from Lovecraft, and invented new Great Old Ones, in an attempt to overuse them so they'd no longer be scary and the fans could go to sleep without their nightlights.
The Great Old Ones are beings of immense power, unbound by the physical laws of this universe. They can do literally anything. Which is usually manifested by whispering nonsense in the ear of some human on the verge of insanity so he'll go over the edge and try to sacrifice one of the companions and over-enunciate DOK-TOR and claim that nothing can stop him now.
In addition to being omnipotent, they're also omniscient. They can see into the past, the future, and even alternate possible futures, telling the Doctor that he will become the Valeyard if he continues on his current path, or revealing that he has a dark secret in his past that he won't admit even to himself, or showing the future of Gallifrey in ruins along every possible timeline. The reason they're always trying to take over the Matrix or the Doctor's brain or whatever to get the knowledge that they already have is so obvious that it needs no explanation here.
The comics, being comics, were legally required to create The Grace, beings even greater and older than the Great Old Ones, who can do even more than literally anything (although of course the Beyonder is even greater and older than them, and can do more than just more than literally anything). The Grace are so powerful they can permanently turn The Mattress good, which lasted almost two issues.
NuWho has no time for such silly power creep, and instead of using the Great Old Ones, gave us perfectly sensible monsters like The Beast, and then basically turned the Doctor into a Great Old One, because why not. After Moffat took over, he brought back the Great Intelligence and implied that he really is a Great Old One like the novels said, and proved it by having him use his great intelligence and infinite power to whisper in the ear of people like Miss Kizlet. He (The Great Intelligence, not Moffat) can also possess rats and post on the internet, which makes him as powerful as Crispin Glover. At this rate, Chibnall will probably have Hastur robbing a diner with a knife for crack money.