In his talk at the 1950 ICM in Harvard, Henry Whitehead introduced the idea of algebraic homotopy theory and said
“The ultimate aim of algebraic homotopy is to construct a purely algebraic theory, which is equivalent to homotopy theory in the same sort of way that ‘analytic’ is equivalent to ‘pure’ projective geometry.”
A statement of the aims of ‘algebraic homotopy’ might thus include the following homotopy classification problem (from the same source, J.H.C.Whitehead, (ICM, 1950)):
Classify the homotopy types of polyhedra, $X$, $Y$, $\ldots$ , by algebraic data.
Compute the set of homotopy classes of maps, $[X,Y]$, in terms of the classifying data for $X$, $Y$.
These aims are still valid, but, within the context of these webpages, with the enlargement of the class of objects of study to include many other types of spaces, and ultimately $\infty$-groupoids.
One may summarise them, optimistically, by saying that one searches for a nice “algebraic” category $\mathbf{A}$ together with a functor or functors
and an algebraically defined notion of ‘homotopy’ in $\mathbf{A}$ such that
a) if $X\simeq Y$ in $\mathbf{Spaces}$, then $F(X) \simeq F(Y)$ in $\mathbf A$;
b) if $f \simeq g$ in $\mathbf{Spaces}$, then $F(f)\simeq F(g)$ in $\mathbf A$,
and $F$ induces an equivalence of homotopy categories
(Here $\mathbf{Spaces}$ is a category, perhaps of topological spaces such as polyhedra or CW-complexes, but it may be larger than this and may contain the sort of ‘generalised space’, topos, etc., used in other contexts such as algebraic geometry, and, of course, $\infty$-groupoids.)
Baues has developed an approach to Whitehead’s basic programme using a mix of cofibration categories and categories with a particular type of cylinder functor, that he calls I-categories. These are treated in separate entries. Cofibration categories are very similar to the dual of K.S. Brown’s abstract homotopy theory, as discussed in category of fibrant objects and BrownAHT.
Ronnie Brown’s nonabelian algebraic topology has developed Whitehead’s theory of crossed complexes along the lines suggested by the original papers of Whitehead, but extending that, in particular, using generalisations of van Kampen’s theorem. (This is discussed in detail in the entry: nonabelian algebraic topology.)
Often the objects of study are the same, and there is an enormous interaction between the two areas, but the aims and objectives seem to be different. Perhaps, tentatively, one could say that ‘algebraic homotopy’ is ‘combinatorial homotopical algebra’.