A Lorentzian metric is the rule spacetime uses to measure the interval between two events, and it behaves differently from an ordinary ruler. Instead of always giving a positive distance, it can come out positive, negative, or zero depending on whether the two events could be connected by a slower-than-light signal, a light signal, or nothing at all.
This is what gives spacetime its built-in sense of cause and effect. Events with a negative interval between them are timelike separated, meaning one could influence the other; events with a positive interval are spacelike separated and can never affect each other, no matter how fast a signal travels.
The Lorentzian metric is the mathematical object that makes concepts like the light cone, proper time, and causal order well defined, rather than just informal descriptions.
Mathematicians describe this behavior with something called the metric’s signature, usually written as one plus sign and three minus signs, in contrast to an ordinary Euclidean metric, which is entirely made of plus signs and treats every direction the same way. That single flipped sign, attached to the time direction, is what separates spacetime geometry from the geometry of an ordinary flat room, and it is the reason time behaves so differently from the three spatial dimensions we move through freely. Physicists sometimes describe this signature as the fingerprint of relativity itself: strip away that one flipped sign, and the equations reduce to ordinary geometry with no special role for time and no built-in speed limit for cause and effect.
The simplest possible Lorentzian metric describes flat spacetime with no gravity at all, and is called the Minkowski metric after the mathematician who first formalized it. Every curved, gravity-filled spacetime that general relativity describes still looks like this flat Minkowski metric if you zoom in closely enough on any single point, which is part of why the Lorentzian structure is considered more fundamental than curvature itself.