In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X.
Some major theorems characterize relatively compact subsets, in particular in function spaces.
Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis.
Mahler's compactness theorem in the geometry of numbers characterizes relatively compact subsets in certain non-compact homogeneous spaces (specifically spaces of lattices).
The definition of an almost periodic function F at a conceptual level has to do with the translates of F being a relatively compact set.