This right lifting property is in this context called the homotopy lifting property, because the maps from are understood as homotopies. In more detail, for every space , any homotopy , and a continuous map , there is a homotopy such that and :
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 where 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.