Causal System Definition
A causal system is one whose output at some current time $ t_0 $ only depends on some input, $ x(t) $, at the current $ t_0 $ or a $ t $ in the past, never from $ x(t) $ at a future $ t $.
Non-Causal System Definition
A non-causal system is one whose output at some time $ t_0 $ depends on future inputs as well as possibly past or current inputs.
EDIT: Added $ t_0 $ per discussion suggestions.
