I don't have anything approaching a complete answer to your question, but I can offer some further hints, some of which you may already know...
Every group of prime order is cyclic (hence abelian), so we can rule out those. Also every group of order p^2, p a prime, is abelian (like the Klein-4 group for instance). We also have:
If p < q are distinct primes, then
1) if p does not divide q-1, then every group of order pq is cyclic (there is only one group of order 15=3*5 for instance)
2) if p divides q-1, then there are exactly two groups of order pq, one cyclic and one non-abelian (6 = 2*3, as you mentioned)
And that is about as far as I can go off the top of my head. This is about as far as Hungerford goes in his beginning graduate text "Algebra". I am sure there are more theorems like these, but i would be very surprised to find that all orders of non-abelian groups have been classified (not that it hasn't, just that I would be surprised ). It's certainly a worthwhile question nonetheless.