"Interpolation" is a broad term for starting with two concrete bounds on a mathematical object, and producing a range of other bounds that continuously fill in "the abstract space of bounds" between the extreme points that you started with. A very typical example is provided by linear operators on the set of functions on a measure space. If $T$ is a linear map defined on simple functions, then there are all sorts of continuity properties $T$ might have, depending on which $L^p$ norm I pick for its domain and which $L^p$ norm I pick for its image. We can talk about the norm $$\lVert T \rVert_{p \to q} = \sup_{\lVert f \rVert_p = 1} \lVert T f \rVert_q,$$ and ask whether this is finite for a given choice of $p$ and $q,$ or more concretely compute this norm as a function of $(p,q).$ Interpolation essentially says: supposing you know $\lVert T \rVert_{p_1, q_1}$ and $\lVert T \rVert_{p_2, q_2}$, can you deduce an upper bound for a family of norms $\lVert T \rVert_{p_t, ...