The Keeper of the Tower is an entity that resided (presumably) in a tower.
Angela's Apprenticeship[]
Angela was an apprentice to The Keeper for some time. During Angela's apprenticeship, he taught her from books kept in The Library. Angela was not allowed in the Library, but one day she snuck inside to look for a hidden drawer. She found the drawer but became trapped inside. Nothing more is known about The Keeper, except that he is male.
Speculation[]
It is possible that The Keeper of the Tower is Tenga. Angela mentions that she was apprenticed to both Tenga and the Keeper at different points. In addition, the 'tower' that the Keeper resides in could be Edur Ithindra, Tenga's dwelling place. Angela is known to heavily dislike both characters. If Tenga and the Keeper are the same person, then it seems that he has lost much of his power in the time between Angela first meeting him and when Eragon meets him. It's also speculation that Angela is the soothsayer and that tower could be where Angela stayed with Galbatorix which is why she doesn't want to hear his name and won't speak about him.