I am having trouble understanding the definition of cofibration as given in Bredon's "Topology and Geometry." In particular, he defines a cofibration using the following diagram:
I am not sure how to interpret what Bredon means by "filling in" the diagram. I assume that the top and bottom horizontal maps are inclusions, but what are the two non-dotted diagonal maps $X\times \{0\}\to Y$ and $A\times I \to Y$? I'm unsure what is given and what we are trying to "fill in". How does one interpret this diagram in terms of homotopies and homotopy extension?
Regarding your questions, both $\phi: X\times\{ 0\} \rightarrow Y$ and $\psi: A\times I \rightarrow Y$ are assumed to be arbitrary (continuous) functions to $Y$. Nevertheless, in order for the diagram to commute we should have that $\phi\circ f\times 1 = \psi\circ \iota$, which ultimately means that $\phi$ and $\psi$ are compatible under $f$. Then, the information you have previously on hand is how to map $X$ into $Y$ (given by $\phi$), and you have a homotopy from $A$ to $Y$ (given by $\psi$), as long as the diagram commutes.
A cofibration $f:A\rightarrow X$ ensures you can "fill in" a homotopy between $X$ and $Y$ (represented in the diagram by the dotted map). The commutativity of the diagram implies that this homotopy maps $X\times\{0\}$ to $\phi(X\times\{0\})$ and that it is equivalent to $\psi$ when composed with $f\times 1.$
In summary, a cofibration $f:A\rightarrow X$ allows you to construct a new homotopy between $X$ and $Y$ such that the first end is a given continuous function ($\phi$) compatible with a homotopy $\psi$ between $A$ and $Y$. In other words, you can use $f$ to "map" homotopies from $A$ to $Y$ to homotopies from $X$ to $Y$, for any space $Y$.
Note: As a rule of thumb, when a textbook says something like "for any space $Y$" and does not comment on the maps that go from (or to) it, the definition covers any possible continuous map, and the diagram should provide any other rule these maps must satisfy.