What, precisely, does Klein's Erlangen Program state? People write that the Erlangen Program is a "program" (like the "Langlands Program"), i.e. a series of related conjectures, which in this case were all solved. There are various intuitive accounts, describing that this program is about relating algebra and geometry, about relating transformation groups of spaces (Lie groups) and different geometries, invariants, etc.
What I haven't been able to find is a precise statement in modern language (e.g. not that of his original paper) of what Klein's conjectures were. What precisely were his conjectures, or equivalently, what results constitute their resolution? Or was there never truly a precise statement?
 A: A small complement:
It is good to have a look at what the contemporaries thought about Klein's program
and how it articulated with the big deal of the time: classical invariant theory.
In particular Franz Meyer's review mentioned in my answer to MO96140 is a useful reference in this regard (Klein's program is mentioned on pages 19, 42 and 45).
At the opposite extreme of the time axis, one can note that Klein's program continues to be a source of inspiration for mathematicians, see e.g. this article posted today on arXiv by Freed and Hopkins. 
A: D. E. Rowe notes that one shouldn't read too much in program:

Klein wrote it to fulfill a formal requirement incumbant upon every new Ordinarius who entered the Erlangen faculty. Its name, in fact, had nothing to do with a new program for mathematical research, but rather derives from having been a “Programm zum Eintritt in die philosophische Fakultät und den Senat” at Erlangen.

(Likewise W. Killing and many others.)
A: For the historical part of the question, what about reading the (professional) historians of mathematics, for instance, as a point of departure : Jeremy Gray, Felix Klein's Erlangen programme, in Landmark Writings in Western Mathematics, ed. I. Grattan-Guinness, Elsevier, p. 544-552, 2005, which explains the circumstances of the paper and its main contents, and gives a bibliography.
Indeed, the programme was not a series of conjectures ! It states a view of geometry in which a geometry was associated to a group of transformations (not uniquely, of course), and (a part often forgotten) this also should provide explicit invariants. It was important because 1) geometry was still currently understood as "geometry" and not "geometries" at the time, 2) it allows to classify them and show the analogies/identity between different geometries (some quite bizarre to-day). As said by others above, from our point of view, a lot was not included, of course, for instance, Riemannian geometry was explicitely out of it (it was of the achievements of the Elie Cartan generation, and specially Elie Cartan himself, to integrate Riemann to this picture). The influence of the programme has been also studied extensively. Best, C. Goldstein
A: Note: I am not sure if you have seen this paper, but I post it here as an answer and I quote a couple of paragraphs, they are not my words! There is also this book.
The first section of Klein's Erlangen Program (E.P.) announced its main theme as follows (E.P., 67): "Given a manifold and a transformation group acting on it, to investigate those properties of figures [Gebilde] on that manifold which are invariant under [all] transformations of that group." In today's language, Klein proposed studying the concept of a homogeneous manifold: a structure $[M,G]$ consisting of a manifold $M$ and a group $G$ acting transitively on $M$. This contrasts sharply with Riemann's concept of a structure $[M;d]$ consisting of a manifold on which a metric $d(p,q)$ is defined by a local distance differential $ds^2 = \sum g_{ij}dx_idx_j$.
Two paragraphs later, Klein restated his proposal in a single terse sentence: "Given a manifold, and a transformation group acting on it, to study its invariants.'' Thus Klein was also proposing to apply to
geometry the concept of an 'invariant' that Clebsch, Jordan, and their
predecessors had previously applied to algebra, and there only to the full
linear group.
A: (Rewritten in response to David Corfield's comment below.) 
A somewhat more modern take on the Erlanger Programm was given in Tarski's 1966 talk What Are Logical Notions? (published 1986), as described in the Wikipedia article on Tarski, which proposes a distinction between what is logical and what is non-logical. The idea that as one loosens the theory (say from Euclidean geometry to affine geometry to topology to...), the relevant automorphism group becomes larger and larger, so that maximal automorphism groups (symmetric groups) correspond to theories of maximal looseness, where one is left with purely logical notions. 
However, it should be said that Tarski's idea was clearly anticipated by F.I. Mautner, writing in 1946; see here. For some commentary on this, see this post by David Corfield at the $n$-Category Café. 
As shameless self-promotion, I'll mention that James Dolan and I dabbled a little in this as well; some results were described at the $n$-Category Café, here and here. There we give describe a Galois correspondence between subgroups of symmetric groups and complete theories, in categorical terms. 
