I know a continuous map $f:X\to Y$ between topological spaces is a weak homotopic equivalence if it induces isomorphisms on the corresponding homotopy groups, but what kind of information do I get (about X and Y) knowing that such a map f exists?
The big theorem about weak homotopy equivalences is Whitehead's Theorem. If $X$ and $Y$ are pointed CW-complexes, and there exists a pointed weak homotopy equivalence $f\colon X\rightarrow Y$, then $f$ is a homotopy equivalence between $X$ and $Y$. Probably the simplest corollary of this theorem is that every CW complex with trivial homotopy groups at all levels is homotopy equivalent to a point.
The notion of a weak homotopy equivalence is eventually generalised to a special kind of arrow (still called a weak equivalence) in a Model category which in some ways generalises homotopy theory to arbitrary categories.