A group $G$ is relatively hyperbolic relative to a collection of subgroups $\mathcal{G}$, if $G$ admits an action on a connected graph, $K$, with the following properties:
(1) $K$ is hyperbolic, and each edge of $K$ is contained in only finitely many circuits of length $n$ for any given integer, $n$,
(2) there are finitely many $G$-orbits of edges, and each edge stabiliser is finite,
(3) the elements of $\mathcal{G}$ are precisely the infinite vertex stabilisers of $K$, and
(4) every element of $\mathcal{G}$ is finitely generated.
This is taken from Bowditch. There are other equivalent definitions, and an inequivalent definition (though groups satisfying the latter are now usually called weakly relatively hyperbolic).
I have two questions about relatively hyperbolic groups. They sound elementary, but I haven't seen an explicit answer in, for example, papers by Farb or Bowditch.
Is a free product of finitely many relatively hyperbolic groups itself relatively hyperbolic (relative to the collection of given parabolic subgroups)?
If $G$ is hyperbolic relative to non-trivial subgroups $H_1,\ldots,H_n$, and each $H_i$ is a subgroup of $H_i'$, is $G$ hyperbolic relative to $H_1',\ldots H_n'$? (Assuming that $H_i'\cap H_j'$ is trivial for $i\neq j$.)
The answer to question 1 is "yes", by results of Osin. Indeed one has, under some mild assumptions, an analogous statement for HNN- Extensions and free products with amalgamation. However for free products these apply trivially.
See "F. Dahmani. Combination of convergence groups" and especially "D.V. Osin, Relative Dehn functions of amalgamated products and HNN-Extensions".
Question 2 has been treated in the comments (@Derek Holt).