A random oracle is an idealization of a hash function $H$: if hash functions were perfect they would be random oracles. This is why it is always easier to consider a hash function a random oracle when one proves something about a larger scheme. Those are "proofs in the random oracle model". [1]
That being said it is still possible to prove things using different, weaker, assumptions about the hash function or even the compression function in the case of a MD hash function (collision resistance for example). Those are "proofs in the standard model"
Is it possible to go from one to the other? Not always since there are separations: there exists schemes that can be proven secure in the RO model that will become insecure as soon as you instanciate the RO by a real world hash function. [2]
Is it the end of the world? Not really, since we also have strong results that use the notion of Indifferentiability from RO: simply put, if your hash function is indifferentiable from a RO then you can replace your RO by your hash function and the scheme will remain secure. [3]
There are (as always) subtleties and what I've just said is not always true but this is good enough to make us feel more comfortable with our assumptions. [4]
Is this RO-indifferentiability just theoretical crypto stuff? No! I haven't checked for all of them but I know that Skein and Keccak 2 of the SHA-3 finalist come with proofs regarding that very property. I believe all of the finalist do. [5] [6]
If you want you can start by reading the following: