One of the main problems in the Langlands program is functoriality, which is roughly speaking: A "map", say $G \mapsto G'$, between groups should "transfer" certain representations.
I will remain pretty vague what "map" and "transfer" are, simply because I only understand special cases of this. The representations are (cuspidal) automorphic representations. Essentially you want to show that automorphic forms are a category with the transforms being the morphisms.
I will describe the Jacquet-Langlands correspondence. A classical way to proof such transfers is via the trace formula. Roughly speaking, it is an identity between the representations and its conjugacy classes. It suffices for proving the transfer that the terms involving the conjugacy classes of $G$ and $G'$ coincide in some sense. Now, these terms are (weighted) orbital integrals associated to each conjugacy class.
Another part of Langlands program is the correspondence with Galois representations, which is probably what you mean with quadratic reciprocity in this context. Here, the strategy is similar only that given a Galois representation, you have to let it act on some geometry which is comparable with the conjugacy classes of $G$. I think that was the key ingredient in Laurent Lafforgue's proof for $GL(n)$ over a global function field.
So the trace formula is the standard strategy for both correspondence and functoriality. It is useful to have a good understanding of orbital integrals therefore.
Now let me address stability: Quote from Harris http://www.institut.math.jussieu.fr/projets/fa/bpFiles/Intro_Harris.pdf, Section 3, which is by the way an excellent survey.
Semi-simple conjugacy classes
in inner forms of
$GL(n)$ are completely characterized by their characteristic polynomials. Similar characterizations are possible for other
groups, but only over an
algebraically closed field.
Now, global fields are never algebraically closed and to overcome the issues involved in that matter is precisely the problem of stability, I think.