An adjunction space is a topological space $Y= B \cup _f X$ determined by an inclusion $i : A \to X$ and a map $f: A \to B$; it can be described briefly as the pushout of the two maps $f,i$. The advantage of this notion is that it is behind the J.H.C. Whitehead notion of CW-complex, which is about inductively attaching cells $E^n$ by means of maps $S^{n-1} \to X_{n-1}$. Good properties of $A,X,B$ are often carried over to $Y$ (eg being Hausdorff). The pushout property allows one to construct continuous functions on $Y$. { cf Topology and Groupoids).
The earliest paper I have on this notion is by Whitehead: "Note on a Theorem due to Borsuk", Bull AMS (1948) 1125-1137. I would be grateful for information on earlier reference to this notion.
Since I was asked to state an official answer, here it is, although it's just copy paste from Engelking's book:
According to Engelking, General Topology, Historical and bibliographic notes of section 2.4, adjunction spaces were defined by Borsuk in 1935 (for compact metric spaces)