Let G be a locally-compact group acting on a locally-compact Hausdorff topological space X (these additional conditions on G and X being certainly satisfied if e.g. G is finitely-generated and X is a proper metric space.)
We say the action is properly discontinuous if for any compact set , for at most finitely many . (cf. the action of on by translations, which is discontinuous: for any [sufficiently small] compact set K, for every ; here we allow finite overlap even at arbitrarily small resolution: this allows for finite stabilizers, parabolic elements, etc.)
In particular this implies that every point has an open neighborhood U s.t. for at most finitely many . (Take U to sit inside some compact set K.)
Note that the second condition is strictly weaker than the first: consider the -action on the punctured plane given by . This satisfies the second condition, but not the first—take K to be any closed annulus centered about the origin, for instance.
The main point of imposing these mildly technical conditions when we study “geometric” group actions is this:
This second condition is equivalent to the quotient map being a covering map. The first condition (the definition) serves to ensure that the map given by is proper, and hence that the quotient X/G is Hausdorff.