The chemical structure of a diamond is defined as an endless lattice in which each carbon atom is covalently bonded to four other carbon atoms situated at the four ends of a tetrahedron. But of course they aren't endless; diamonds have edges and faces. So what happens when the lattice reaches the edge and there aren't any other carbon atoms to bond to? What is the "surface layer" of a diamond composed of?
This is a nice question. Although I do not speak with authority on diamonds, most crystal surfaces have several imperfections which mitigate the hanging valencies that would otherwise appear for the atoms (constituents in general) at the surface.
Impurities. This is the primary method which helps satisfy the unfulfilled valencies at the surface. Many metallic, non-metallic or compound impurities are usually incorporated throughout the crystal, but are usually high at the surface. By bonding with these excess impurities, the surface carbon atoms can fulfill their valencies. An extremely pure crystal surfaces (lots of "hanging" valencies) will quickly adsorb most materials that it comes in contact with. Chemisorption or physisorption depends on the nature of the crystal, metals prefer physisorption while non-metals have at least unimolecular chemisorption usually.
Another way to mitigate surface valencies is distortion. The surface layer might not follow the exact space-filling Bravais lattice structure and may form short range distortions in structure to try to make the unfulfilled valencies satisfied. There might be kinks, depressions, steps etc which at least help in completing the valencies of a few atoms (at the boundary of distortion, say, a "step" structure).
If none of these operate, there is an inherent surface energy or surface tension between any two phases which quantifies the desire of the surface atoms to fulfill their hanging valencies to resemble the atoms in bulk which are stably attracted. In case of metallic and ionic crystals, this is comparatively low, but may be high in a covalent crystal like diamond.
In all cases, the surface layer of constituents (actually first few layers) are different from the atoms in bulk in energy, stability, forces acting (unbalanced on the surface and balanced in the bulk) and their structure/orientation.
At the surface of a diamond, the carbon atoms are generally bound to various other atoms (or groups of atoms) with lower valence.
In principle, pretty much anything that could act as a substituent group in organic chemistry could also be bonded to the surface of diamond, but in practice, you're most likely to see simple groups with low reactivity, mostly either plain hydrogen or various oxygen-containing groups (I suspect mainly hydroxyls and/or ketones).
According to Wikipedia (which itself cites Harlow 1998, The nature of diamonds):
"Naturally occurring diamonds have a surface with less than a half monolayer coverage of oxygen, the balance being hydrogen."
Although the diamond lattice itself is very stable and inert, these surface groups can participate in chemical reactions, and indeed, it's possible to alter the composition of the surface layer by reacting it with appropriate chemicals. For an example, see this random paper I found on Google.