ngeo, I don't know as much physics as Tim and Tensor, but I do understand some of the math. Maybe my input can complemement theirs. If I say something that is wrong, I hope someone corrects me.

The quantities in Tim's post with the "ab" written in parentheses are tensors. Tensors transform in a predictable way whenever the local coordinate system is changed. Thus, although the quantities in a tensor are described in terms of local coordinates, the tensors themselves are independent of the choice of coordinate system. Hence, we know, in a sense, not only how they look from a particular point of view, but also how they would look from any other point of view.

The key quantities on the left are the functions g(ab), where a and b are indices each of which runs from 1 to 4 (because spacetime has 4 dimensions). Hence, there are really 16 of them, although some of them are equal to each other. The g(ab) determine the Riemannian (actually, pseudo-Riemannian) metric at each nonsingular point in spacetime. The metric tells how to measure distances and angles at each point. Thus, for instance, it can be used to calculate the length of each path and to determine which paths are geodesics, a geodesic being the shortest path between two points.

The metric determines the curvatures, so the R(ab) and R on the left are not independent of the g(ab) and, in fact, can be calculated from the g(ab). Since L is a constant, the entire left side of the equation can be expressed in terms of the g(ab) and their derivatives, but that would look like a huge, complicated mess.

So far, no physics has been introduced. All the terms on the left are geometric. Depending on the values of the g(ab), they could describe any pseudo-Riemannian manifold, whether or not that manifold had any relation to physics or physical reality.

The left side alone is like a subject without a predicate. It is not constrained. Setting it equal to the right side creates a statement which relates that subject to the universe. The equating of the two sides incorporates the g(ab) into a set of equations, which constrain the g(ab). Now they must conform to the laws of physics. These constraints (Einstein equations) determine the g(ab), which , in turn, determine the curvatures and, hence, the shape of spacetime.