I agree with your definition of locality (probably not surprising :)).
Causality I would say is the statement that an event in the future should not affect an event in the past. We can formulate this in classical physics terms. Causality is necessary in order for there to be a well defined initial value problem: I should be able to choose an initial time slice, specify the field values and derivatives on that slice, and evolve the system forward from there unambiguously. Acausality would allow an event from the future to come back and affect what's going on in the past--in principle that would allow the field evolution to change the initial conditions you started with.
If you like, causality is the requirement that there should be no time machines that allow me to send information into my past--I should not be allowed to kill my own grandfather.
If you don't demand Lorentz invariance, locality and causality are distinct concepts. I can certainly imagine non-local theories that are causal--Newton's action at a distance version of gravity is certainly causal, but it is nonlocal. Similarly, I can imagine a universe where I can press a button and reverse the flow of time for me (ie, my clock runs in the opposite direction of the rest of the universe), where I can only interact with things locally but I now have clearly violated causality.
These notions however become related once you demand Lorentz invariance. The reason is that the notion of simultaneity is relative. In particular, the time ordering of spacelike separated events becomes observer dependent. So if two spacelike separated events can affect each other (which is definitely non-local), there is a frame where I am using this spacelike communication to talk to someone in my past. She can then (provided that she can also perform spacelike communication) talk to someone in my past but also inside my past light cone. So you can create a loop of communication that ends up in my past light cone. In this example, no one is moving faster than light (or, maybe more accurately, the non-local communication allows for superluminal transfer of information), but the nonlocal transfer of information has allowed something I said now to end up in my past light cone.
So if we don't want to allow spacelike transfer of information, what can we do? Well at a fixed time the only event that is not spacelike separated from me is the event where I am located. So I can only affect the fields and their derivatives at my location.
As a warning, in gravity when the spacetime metric becomes dynamical, all of this becomes more complicated! In special relativity when the metric is fixed, things are more clear.
This post imported from StackExchange Physics at 2015-11-01 19:28 (UTC), posted by SE-user Andrew