The Great Intelligence is a member of a race of beings, known as the Great Old Ones, that existed in the Universe prior to our own. It's originally known as Yog-Sothoth, and they escaped the destruction of their own universe by going to a parallel universe that ended a second after theirs, and then shifted once more into the new universe (ours). It had no physical form in our universe for a while and was originally described as a cloud.
The Great Intelligence - Tardis Data Core
Originating from the universe before this one, Yog-Sothoth and his
brethren survived the end of their universe by passing through a
parallel universe/dimension that ended one second after theirs.
Shifting again allowed them to enter the current universe shortly
after it began expanding. (PROSE: All-Consuming Fire, Millennial
Rites) Like the other Great Old Ones, it may have lost its body when
it ran from the Fendahl. (PROSE: White Darkness) The Intelligence
constantly sought physical existence to replace being a shapeless,
formless cloud hanging in space. (TV: The Web of Fear)
It took a few forms throughout the years...
A pyramid of control spheres in the Homevid Downtime and The Abominable Snowmen..
When not using a living being, it maintained a basic manifestation as
a three-sided pyramid composed of control spheres (TV: The Abominable
Snowmen) or ivory. (HOMEVID: Downtime)
Living Snow using low level telepathic field in Abominable Snowmen..
Then after getting enough mental energy and feeding for a while after Bells of Saint John he formed his own body via host of beings known as The Whisper Men though it wasn't complete physical manifestation.
The Whisper Men could morph into the appearance of Walter Simeon at
will, and much like the human embodiment of the Great Intelligence,
seemed to be hollow on the inside, or unreal. They moved slowly, and
lacked features, except for extremely white faces with blackened skin
around their mouths and sharp teeth. They had angular faces with vague
depressions where eyes should be, and exaggerated chins and cheeks.
They wore the same Victorian garb as the Great Intelligence's form as
Walter Simeon. (TV: The Name of the Doctor)