The pictures you typically see of orbitals in textbooks or online (a sphere for an s orbital, a kind of 3D figure of 8 for a p orbital etc) are representations of a mathematical function called the wavefunction. This is a 3D function that describes the wave-like properties of an electron.
Where do these wavefunctions come from? They are solutions to the Schrödinger equation, which is the central equation in quantum physics. In the case of orbitals we are thinking about electrons bound in atoms. For bound electrons only certain functions are valid solutions for the Schrödinger equation, and so only certain electron orbitals are "allowed" in atoms. You may know the names of these solutions already, the first few are commonly called 1s, 2s, 2p, 3s, 3p, 3d, 4s etc etc and so on forever.
Each of these orbital labels refers to a mathematical function (a wavefunction) which is a valid solution to the Schrödinger equation. It is then a simple matter to describe how orbitals can combine (hybridize, overlap, whatever you want to call it) - we just add the wavefunctions together. So the chemical bond in H2
can be described as a sum of the two H 1s orbital wavefunctions.
In a sense then, the answer to your question is that these are mathematical functions which come from the prevailing theory (quantum mechanics). So that's all nice and neat then.