CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Serre fibration $\Leftarrow$ Hurewicz fibration $\Rightarrow$ Dold fibration $\Leftarrow$ shrinkable map
A continuous map $p \;\colon\; E\longrightarrow B$ of topological space is called a Hurewicz fibration it it satisfies the right lifting property with respect to maps of the form $\sigma_0 \;\colon\; X\cong X\times\{0\}\hookrightarrow X\times I$ for all topological spaces $X$.
This right lifting property is in this context called the homotopy lifting property, because the maps from $X\times I$ are understood as homotopies. In more detail, for every space $X$, any homotopy $F:X\times I\to B$, and a continuous map $f:X\to E$, there is a homotopy $\tilde{F}:X\times I\to E$ such that $f =\tilde{F} \circ\sigma_0 :=\tilde{F}_0$ and $F=p\circ\tilde{F}$:
Strictly speaking, the “all” in this context should be interpreted to refer to all spaces in whatever ambient category of spaces one is working in, since frequently this is a convenient category of spaces. In theory, therefore, a map in such a category could be a Hurewicz fibration in that category without necessarily being a Hurewicz fibration in the category of all topological spaces, but in practice this usually doesn’t make a whole lot of difference.
Instead of checking the homotopy lifting property, one can instead solve a universal problem:
A map is a Hurewicz fibration precisely if it admits a Hurewicz connection. (See there for details.)
There is a Quillen model category structure on Top where fibrations are Hurewicz fibrations, cofibrations are closed Hurewicz cofibrations and weak equivalences are homotopy equivalences; see model structure on topological spaces and Strøm's model category. There is a version of Hurewicz fibrations for pointed spaces, as well as in the slice category $Top/B_0$ where $B_0$ is a fixed base.
The historical paper of Hurewicz is
A decent review of Hurewicz fibrations, Hurewicz connections and related issues isin
A textbook account of the homotopy lifting property is for instance in
See also the textbooks on algebraic topology by Whitehead and Spanier.