Here is my top ten list of aliens. I have picked them to be all different kinds of things. Only one I think is an ordinary corporeal respirating animal:
Solaris: a planet-large entity, from the book of the same name by Stanislaw Lem. Runner up in this category - the planet-entity in Le Guin's Vaster than Empires and More Slow.
The Tines: from A Fire Upon the Deep by Vernor Vinge. Each individual is a small pack of dog-like creatures who communicate with ultrasonics. Runner up in the group-entity category - the Braids, sentient colonies of centipedes, in Anvil of Stars by Greg Bear.
The Thing from Who Goes There by John W Campbell and the Carpenter Film. Best-ever shape shifting entity. Runner-up, the humanoid shape shifter species in Iain M Banks' Consider Phlebas.
The Elder Gods from the works of HP Lovecraft. Best pitiless godlike entities from Other Realms. Runner up, whatever lurks in Pratchett's Dungeon Dimensions. Better not to know.
The Greys (or generally the mythical UFO aliens). An alien species invented by no person - by everyone. They exist as mental projections within the human subconscious I suppose. The best story I know in which they feature is Miracle Visitors by Ian Watson.
The Alien: a genetically engineered species created as a bioweapon. Runner up in the bioweapon category - the polymorph from Red Dwarf ('Genetically modified life forms - No thanks')
Pierson's Puppeteers from Ringworld by Larry Niven (suggested by houseboatonstyx). A good example of a non-humanoid but biologically conventional alien species. By 'conventional' I mean they are corporeal respiring carbon based life forms, that are born and die.
Shai-Hulud or Sandworm, from the Dune books. Another biological species, but one which derives energy from electrostatic forces.
The Qax (from Stephen Baxter's Xeelee novels) as an example of an alien species which has a physical manifestation but not as a conventional body. The Qax minds are composed of convection currents in a substrate - originally mud I think, then gas and finally quantum matter. Basically they are intelligent mud puddles.
And finally the white mice from Hitch-hikers' Guide to the Galaxy, which are the extrusion into our universe of vast multi-dimensional hyperbeings. Who incidentally programmed Deep Thought, who discovered the meaning of life. Not bad for mice.