A light cone marks the boundary of everything a single event in spacetime could possibly affect or be affected by. It gets its name because if you plot every direction light could travel outward from that event over time, the result forms a cone-shaped surface.
Everything inside the future light cone is a place that event could still influence, since a signal from it could arrive there without exceeding the speed of light. Everything inside the past light cone could have influenced that event. Everything outside both cones is causally out of reach entirely.
Light cones replace the old idea of instantaneous, universal cause and effect. In relativity, influence can only spread at or below the speed of light, and the light cone is the geometric picture of exactly how far it can reach.