From Prof. John Baez (I asked him for a quick note):
There are many different theories of space, so there are many different answers to the question “what is space?” But if you ask about the role played by space in general relativity, I’d say that Einstein discovered that space is not so important: we should instead think about spacetime, because this is something that different observers can agree on. His theory of spacetime says that it is a differentiable manifold with a Lorentzian metric. So, it’s a kind of entity, with very definite properties, that are nicely described using math.
Really explaining what I just said would take about a year or two! I don’t want to spend my time explaining general relativity anymore, since I wrote a book about it, and an expository paper.