Basically a beginner type of topology question here, but I am trying to understand something and am a bit stuck on a definition.
According to J.P. May, a fibration is a map $p : E \to B$ such that for all spaces $Y$, that embed into $E$ by some map $f$, and have a compatible homotopy onto $B$ by $h: Y \times I \to B$, where $h(y, 0) = p(f(y))$ for all $y\in Y$; there is a unique extension of $h$ to a homotopy $\tilde{h} : Y \times I \to E$ such that, $p(\tilde{h}(y, t)) = h(y,t)$.
Now what I am trying to wrap my head around is what that actually means. What restrictions does this place on the space $E$ and the maps $p$? Are there examples of continuous maps $p$ which do not satisfy this property? If so, how is this supposed to generalize the usual notion of a fiber bundle, and what are the fibers constrained to be? Can the fibers be different dimensional spaces or have varying genus?