The metric tensor is the mathematical object that tells you how to measure distances and time intervals at every point in spacetime. It is the precise version of a Lorentzian metric, written so it can vary smoothly from place to place.
In flat, gravity-free spacetime, the metric tensor is simple and constant everywhere. Near a massive object like a star, it changes from point to point, and it is exactly this point-to-point variation that general relativity identifies as the curvature of spacetime.
Solving Einstein’s field equations means finding the metric tensor for a given distribution of matter and energy, since once you know it, you know the geometry of spacetime itself, and therefore how objects will move through it.