They appear starlike, but, in fact, are objects in extragalactic space, i.e. outside of our Galaxy, and they are very far away.
Quasars are the most distant and luminous objects in the visible universe.
In the cores of these objects, there is something which produces a tremendous amount of energy, much more energy than is produced by all of the stars in a typical galaxy. It is not entirely clear what the source of this tremendous energy is, but most likely, it is a very big black hole.