The Geometric Langlands Conjecture concerns an equivalence of two particular DG categories. The notion of DG category is an enhancement of the notion of triangulated category (abelian categories will not be sufficient). An equivalent theory is provided by the notion of stable infinity-category, tensored over k (k is our ground field, assumed of characteristic 0).
It may be a bit tricky to find a source that explains how the theory is set up. But you should not necessarily care about the technical details. Below are some suggestions for possible sources.
A summary of what one needs to know about DG categories is in Sections 1, 2 & 3 of Gaitsgory's notes and repeated in Section 1 of Drinfeld-Gaitsgory
For a more systematic treatment, try Toen's "Lectures on dg-catgories" in "Topics in algebraic and topological K-theory", LNM 2008.
An excellent (and readable) treatment of stable infinity-categories is Chapter 1 of Lurie's book on Higher Algebra.