By $Ho(Top)$ one denotes the category which is the homotopy category of Top with respect to weak equivalences given
either by homotopy equivalences – $Ho(Top)_{he}$.
or by weak homotopy equivalences – $Ho(Top)_{whe}$.
Depending on context here Top contains all topological spaces or is some subcategory of nice topological spaces.
The study of $Ho(Top)$ was the motivating example of homotopy theory. Often $Ho(Top)$ is called the homotopy category.
The simplicial localization of Top at the weak homotopy equivalences yields the (∞,1)-category of ∞-groupoids/homotopy types.
Let now $Top$ denote concretely the category of compactly generated weakly Hausdorff spaces. And Let $CW$ be the subcategory on CW-complexes. We have $Ho(CW)_{whe} = Ho(CW)_{he} = Ho(CW)$.
There is a functor
that sends each topological space to a weakly homotopy equivalent CW-complex.
By the homotopy hypothesis-theorem $Ho(CW)$ is equivalent for instance to the homotopy category $Ho(sSet_{Quillen})$ of the standard model structure on simplicial sets.
The category $Ho(Top)_{he}$ can be studied by testing its objects with objects from $Ho(CW)$. This is the topic of shape theory.