Let $R$ be the ring of integers in an algebraic number field. There are beautiful descriptions of $K_0(R)$ and $K_1(R)$. Namely, $\tilde{K}_0(R)$ is the class group of $R$ and $K_1(R)$ is the group of units of $R$. Question : Is there a nice description of $K_2(R)$ (or at least some reasonable conjectures)? I couldn't find much about this in Milnor's or Rosenberg's books on algebraic Ktheory, so I expect that the answer is pretty complicated. Is it maybe at least known in some special cases (say, for $R$ the integers in a quadratic extension of $\mathbb{Q}$)?

It's a theorem of Garland that $K_2(R)$ is finite. Perhaps the best way to get a handle on it is to use Quillen's localization sequence $$0\rightarrow K_2(R)\rightarrow K_2(F)\stackrel{T}{\rightarrow} \oplus_v k(v)^*\rightarrow 0,$$ where $F$ is the fraction field and the $k(v)$ are the residue fields. The map $T$ is the sum of the tame symbols, which is surjective by a theorem of Matsumoto. The injectivity on the left follows from the vanishing of $K_2$ for finite fields. This isn't much of an answer, but considering $K_2(R)$ as a subgroup of $K_2(F)$ seems a reasonable way to start some concrete considerations. For a detailed discussion of an algorithm that proceeds essentially along these lines ('Tate's method), see the paper Belabas, Karim; Gangl, Herbert Generators and relations for $K_2( O_F)$. $K$Theory 31 (2004), no. 3, 195231. Added, 8 July: I'm sure most people know this, but I forgot to mention (for newcomers) the fact that $$K_2(F) = F^\times\otimes_{\mathbf Z} F^\times/\langle a\otimes(1a)\mid a\not=0,1\rangle,$$ which I suppose motivates the original question, and makes it worthwhile to view $K_2(R)$ as a subgroup. 


I think Kolster's survey is a good introduction to questions related to arithmetic interpretations of higher Kgroups. 


For the conjectural order, see the BirchTate conjecture http://eom.springer.de/B/b110560.htm . 


$K_1$ being the units, and Dirichlet's theorem on the unit group, generalize to odd Kgroups via regulator maps and related conjectures and theorems (starting from work of Borel, Bloch, Beilinson, Zagier, and many others). For the even Kgroups I don't know of any such description. Charles Weibel's web page has a Ktheory textbook with a chapter on $K_1$ and $K_2$, and a paper on the early history of Ktheory. Both of those discuss $K_2$ of rings of integers in a number field, but I didn't see any characterization analogous to the one with regulators for $K_1$ or $K_3$. I'm certainly not an expert on this subject but I have the feeling that if a snazzy description of $K_2({O_F})$ were proven or conjectured, it would be widely known. 


I've never heard of $K_2(R)$ having a description as a more easily/elementarily described object attached to $R$. Borel showed that $K_2(R)$ is torsion. The order of $K_2(R)$ appears in "Theorem" 31 of Soulé's notes on higher algebraic Ktheory of rings of algebraic integers where $K_2(R)$ (I think that "Theorem" 31 is called the QuillenLichtenbaum conjecture). 

