Owls are the sometimes deity, sometimes fowl, often puzzling force lurking among most of the curious history of FEZ. They are definitely messengers. Owls were able to experience the 3rd dimension way before the Visitor's arrival or the power of the Fez itself was introduced to the Big-heads, thanks to the owls' ability to rotate their heads. It is implied that this has given them some measure of omniscience, and made them an object of devout worship for the Long-heads. The candles that usually are scattered around owl statues kinda reinforces that.
Aside from the many monuments and effigies to owls throughout the FEZ universe, there are four living owls that Gomez can discover and speak with. Their locations are initially hinted at in Geezer's house, where a large painting is hung on each wall suggesting where the birds may be found. Another hint lies in a room in the under-construction City of Zu, where Dot urges the player to
"HEAR THE OWLS. SEE THEIR EFFIGY. ASSEMBLE THE PARLIAMENT."
The game gives you a tip on where the owls are located. If you go to the mayor's house in Villageville, you have miniatures of the owls and they sitting on specific trees.
THE THIRTEEN CIRCLES. THE 64-BIT NAME OF GOD.
What does it mean? Possibly it is a hint to the monolith puzzle. Nobody is sure yet. There are several concentric circles hidden in the floors of the game, in different rooms. They are all concentric squares (square-circles?) and can be only seen through the first-person view/mode. However, they are not thirteen, but eight. The image below refers to this theory here:
THE MANY ARE ONE. THE ONE IS MANY.
What does it mean? I assume that a single mind, soul or spirit inhabits all the owls through the game. This would make the owls / THE OWL an almost omniscient, omnipresent being.
HEXAHEDRON. OCTAHEDRON. DODECAHEDRON. HER SACRED GEOMETRY.
What does it mean? The therms refer to polyhedra (like polygons, but in 3D) of 6, 8 and 12 faces. Similar to some RPG dice, pictured below. Maybe her (the big Owl) considers these shapes to be sacred, but I am still not sure where these shapes / monuments can be seen in the game.
A POINT OF ORIGIN. AFTER AND BEYOND.
What does it mean? Maybe that all life begins and ends with the owls? Either that the knowledge of things begins with them, and that the first contact that many zuish and non-zuish people had with the owls was after their death, and that's why there are so many owls in the cemetery? Maybe that what's beyond life is also to do with the owls? Just my impression.
If you are serious about the "hearing the owls, finding their effigy (statue) and assembling the parliament", try to collect all owls here!