A CW-pair consists of a CW complex $X$ (with cell decomposition $\mathcal{E}\equiv\{e_\alpha\}_{\alpha\in I}$) together with one of its subcomplexes $A$ (a closed subspace of $X$ consisting of a union of cells in $\mathcal{E}$; this itself forms a CW complex).
In the definition of a generalized cohomology theory in the Wikipedia article Cohomology, it mentions contravariant functors $h^i$ from the category of CW-pairs to the category of abelian groups. But what are the morphisms in the category of CW-pairs?
The Wiki on the Cellular Approximation Theorem says that a 'map' between CW pairs $(X,A)$ and $(Y,B)$ is a (presumably continuous) function $f:X\to Y$ with $f(A)\subseteq B$. But this does not seem like a strong enough condition to preserve the cell structure. For example, there is no mention that cells in $X$ are mapped into unique cells in $Y$, or even a finite or countable union of them. Please enlighten me and I apologise in advance for my naivety.
It is the definition as you quoted: A map between CW pairs $(X,A)$ and $(Y,B)$ is a continuous map $f:X \to Y$ such that $f(A) \subseteq B$.
There is no need for $f$ to be cellular (i.e. preserve the skeletons). In fact, if you wish, you can make $f$ cellular up to homotopy anyhow - this is the statement of the Cellular Approximation Theorem.