In the context of relativity: causality concerns the questions which events can, in principle, cause which other events and which events are two far apart for one to influence the other. In special relativity, nothing, no moving object, no information, no influence can move faster than light. Thus, no event can influence another if the two events happen too far apart for light to travel between from the first event to reach the location of the second event in time. In other words, light propagation determins the causal structure of spacetime (cf. light-cone). Models or theories respecting this causal structure are themselves called causal – an example are the relativistic quantum field theories.
In general relativity, the cosmic speed limit light speed is only defined locally: In a side-by-side race, no object, no influence can overtake a light signal. This, too, leads to a causal structure, to strict rules which event can influence which other event. As gravity deflects and delays light signals, matters are more complicated than in special relativity, but it’s still true that the causal structure is completely determined by how light propagates in the spacetime in question.