For context for Tom's answer,
let me state the naive version of the conjecture, which has been around since around 1997 I think (due to Beilinson-Drinfeld). It calls for an equivalence of (dg) categories
$$D(Bun_G(X))\simeq QC(Loc_{G^\vee}(X))$$
between (quasi)coherent $D$-modules on the stack of $G$-bundles on a curve $X$, and (quasi)coherent sheaves on the (derived) stack of flat $G^\vee$ connections on the curve.
Moreover (and this is where most of the content lies) this equivalence should be an equivalence as module categories for the spherical Hecke category $Rep(G^\vee)$ acting on both sides for every choice of point $x\in X$. The action on the right is given by simple multiplication operators (tensor product with a tautological vector bundle on $Loc_{G^\vee}(X)$ attached to a given representation of $G^\vee$ and point $x$. On the right hand side the action is by convolution (Hecke) functors, associated to modifications of $G$-bundles at $x$ (relative positions at $x$ of $G$-bundles are labeled by the affine Grassmannian, and in order to formulate this statement we use the geometric Satake theorem of Lusztig, Drinfeld, Ginzburg and Mirkovic-Vilonen).
The equivalence can be further fixed uniquely by using "Whittaker normalization",
a geometric analog of the identification of L-functions in the classical Langlands story.
In this form the conjecture is a theorem for $GL_1$, using the extension of the Fourier-Mukai transform for D-modules. There are no other groups for which it's known (yet - though perhaps Dennis already has $SL_2$), and few curves (one can do $P^1$ for example). Also (as Scott points out) this is only the unramified case, and there are natural conjectures to make with at least tame ramification (a "parabolic" version of the above).
There's also a close variant of this conjecture which comes naturally from S-duality for N=4 super-Yang-Mills, thanks to Kapustin-Witten. [Edit: in fact the physics suggests a far more refined version of the whole geometric Langlands program.]
It is a theorem of Beilinson-Drinfeld if we restrict on the left hand side to D-modules generated by D itself, and on the right to coherent sheaves living on opers.
It is also well known that this conjecture as stated is too naive, due to bad "functional analysis" of the categories involved (precisely analogous to the analytic issues appearing eg in the Arthur-Selberg trace formula). On the D-module side, the stack $Bun_G$ is not of finite type, and one might want to make more precise "growth conditions" on D-modules along the Harder-Narasimhan strata. On the coherent side, $Loc$ is a derived stack and singular, and one ought to modify the sheaves allowed at the singular points -- in particular at the most singular point, the trivial local system. [Edit:One approach to correcting this involves the "Arthur $SL_2$" -- roughly speaking looking at local systems with an additional flat $SL_2$-action that controls the reducibility..this comes up really beautifully from the physics in work of Gaiotto-Witten, one of the first points where the physics is clearly "smarter" than the math.] These two issues are very neatly paired by the duality. If one restricts to irreducible local systems
and "cuspidal" D-modules, these issues are completely avoided (though the conjecture even on this locus remains open except for $GL_n$, where I believe it can be deduced from work of Gaitsgory following his joint work with Frenkel-Vilonen, see his ICM).
In any case, Dennis has now given a precise formulation of a conjecture, which was at some point at least on his website and follows the outline Tom explained. It is very clear that homotopical algebra a la Lurie is crucial to any attempt to prove this, and Dennis and Jacob have made (AFAIK) great progress on this. The basic idea of the approach is the same as that carried out by Beilinson-Drinfeld and suggested by (old) conformal field theory (in particular Feigin-Frenkel) and (new) topological field theory (Kapustin-Witten and Lurie) -- i.e. a local-to-global argument, deducing the result from a local equivalence which comes from representation theory of loop algebras. The necessary machinery for "categorical harmonic analysis" is now available, and I'm looking forward to hearing a solution before very long..
Edit: In response to Kevin's comment I wanted to make some very informal remarks about ramification.
First of all one needs to keep in mind the distinction between reciprocity (which is what the above conjecture captures) and functoriality. In the geometric setting the former is strictly stronger than the latter, while in the arithmetic setting most of the emphasis is on the latter. It is in fact quite easy to formulate a functoriality conjecture in the geometric setting with arbitrary ramification. Namely, fix some ramification and look at the category of D-modules on the stack of G-bundles with corresponding level structure. Then this is a module category over coherent sheaves on the stack of $G^\vee$ connection
with poles prescribed by the ramification (eg we can take full level structure and allow arbitrary poles). Then one can conjecture that given a map of L-groups $G^\vee\to H^\vee$ the corresponding automorphic categories are given simply by tensoring the module categories from $QC(Loc_{G^\vee}(X))$ to $QC(Loc_{H^\vee}(X))$ (everything here must be taken on the derived level to make sense). It is not hard to see this follows from any form of reciprocity you can formulate. And there is also a geometric version of the Arthur-Selberg trace formula in the ramified setting (under development).
Second, one can make a reciprocity conjecture with full ramification, though you have todecide to what extent you believe it. In the "completed"/"analytic" form of geometric Langlands that comes out of physics such a conjecture in fact appears in a paper of Witten on Wild Ramification. Roughly speaking in the above reciprocity rather than just looking
at the module category structure for D-modules over QC of local systems, you can ask for them to be equivalent... not stated anywhere since maybe I'm too naive and this is known to be too far from the truth, but I think more likely people haven't thought about it very much.
Third, the local story: of course in the p-adic case Kevin discusses one restricts to $GL_n$. There are two things to point out: first of all in the geometric setting one is interested in all groups, and $GL_n$ is not much simpler as far as our understanding of the local story goes. Second, while everything is much harder and deeper in the p-adic setting than the geometric setting, it's worth pointing out that a formulation of a local geometric Langlands conjecture is a much more subtle proposition, involving the representation theory of loop groups on derived categories which is only beginning to be within the reach of modern technology (even at the level of formulation of the objects!)
That being said, there are rough forms of local geometric Langlands conjectures developed
by Frenkel, Gaitsgory and Lurie. It is best understood in the so-called "quantum geometric Langlands program", a deformation of the above picture involving the representation theory of quantum groups, where Gaitsgory-Lurie give a precise general local conjecture and make progress on its resolution. The usual case above is a bit degenerate and one needs to be more careful. In any case the rough form of the local conjecture is an equivalence of 2-categories (again everything has to be taken in the appropriate derived sense) between "smooth" LG-actions on categories and quasicoherent sheaves of categories over the stack of connections on the punctured disc..
--THAT being said we don't have a proof of this for $GL_n$ so again the number theorists win! just wanted to give some sense that there is a reasonable understanding of full ramification.
