A metric is a generalization of distance, capturing some of its characteristics. It is a function mapping pairs of points in a space to non-negative numbers, such that the metric from a point to itself is zero, that applying it to the reverse (i.e., B to A instead of A to B) yields the same result, and that the metric from A to C is always less than or equal to the metric from A to B plus that of B to C, all statements that would hold true of ordinary distance.
In physics, metrics of non-straight lines are of interest, e.g., specifying "the metric of A to C through B", and for curved lines in continuous spaces, using a line integral.
A variation on the metric concept is relevant to space-time in relativity, i.e., a metric, but allowing a zero metric in some cases where points are distinct. Examples of metrics used in astrophysics:
Special cases or variations on GR (modified GR) are often defined by a formula for its particular metric.