the metric (spacetime metric)
The metric is the mathematical object that tells you the true distance between nearby points and the true time elapsed between nearby events. Written as ds^2, it acts like a local ruler-and-clock combined: hand it two neighboring points and it returns the actual interval between them. Once you know the metric everywhere, you know the entire geometry, and in general relativity that means you know the gravitational field.
Think of a map of a mountainous region. The flat printed map looks uniform, but a metric is the extra rule that says how many real meters each map-centimeter stands for at each spot, including how the terrain stretches and tilts. In curved spacetime the metric does the same job for the four dimensions of space and time, telling you how clocks and rulers behave from place to place.
Far from any mass, the metric reduces to the flat one of special relativity, where time and space combine in the familiar Pythagorean-like rule. Near a star or planet it becomes warped, encoding effects like the slowing of clocks deep in gravity and the bending of light. Solving Einstein's equations means finding the metric for a given distribution of matter and energy.
The flat-spacetime metric; gravity shows up as deviations of g_{μν} from this simple form.
The same physical geometry can be written with very different-looking metrics depending on the coordinates chosen; only coordinate-independent quantities like the interval are physically meaningful.