There are a few terms and ideas that are related here, but not quite the same. You mention two separate things; these two conditions work together to guarantee such things as the Lorentz invariance of the S-matrix.
alemsalem said:
I know that two fields have to commute at space-like separations so that observations don't affect each other at these separation.
Here's the first: I'd call it causality. It basically makes sure that time-ordering is unambiguous and Lorentz invariant. There is no invariant definition of before and after for spacelike separated events, so we make it so that order doesn't matter: operators must commute at spacelike separation.
alemsalem said:
what about the thing where the Lagrangian can't couple fields at different locations and time, is it so that we get a normal (local) differential equation and not one that involves the field at different coordinates.
This is what's normally called locality (in QFT): it's the condition that the action can be written as
S = \int d^4 x \mathsc{L}(x)
and, for example, there can be no terms like
\int d^4 x d^4 y \phi(x)\phi(y).
This isn't actually required to preserve Lorentz invariance, but it's a very convenient way to guarantee it, and is almost always assumed to be true*. As you say, this makes the classical equations of motion into PDEs, which is enough to prevent the classical theory showing action-at-a-distance.
These two things together ensure Lorentz invariance. Weinberg vol. 1 has more details towards the beginning (in the scattering bit I think).
Locality in a broader sense is a bit of a sensitive subject in quantum theories and not an area a know a huge amount about, but a search for such things as entanglement, the EPR paradox and Bell's inequality will give a good overview.
*Nonlocal actions don't appear in standard model physics. But I've seen them show up in effective field theories, and there are probably lots more applications for them in beyond SM physics, condensed matter or other areas.
