The best known explicit Ramsey graph construction is in the paper:
Boaz Barak, Anup Rao, Ronen Shaltiel, Avi Wigderson: 2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction. STOC 2006: 671-680
Call a graph $K$-Ramsey if it doesn't have a $K$-clique or a $K$-independent set. They prove
There is an absolute constant $\alpha > 0$ and an explicit construction of a $2^{2^{\log^{1−\alpha} n}} = 2^{n^{o(1)}}$-Ramsey graph over $2^n$ vertices, for every large enough $n \in {\mathbb N}$.
Here, "explicit construction" means roughly that there is an efficient algorithm which when given the string of $N$ ones, it outputs an $N$-node $K$-Ramsey graph. (I know this is "stronger" than what you would like, but you should still check these things out for fun.)
Before the above paper, the best known explicit construction was by Frankl and Wilson, who showed that there are $2^n$ node graph that are about $2^{\Omega(\sqrt{n})}$-Ramsey. Noga Alon had an alternative construction but I think it only matched Frankl and Wilson. See the above paper for more details.
All these constructions are very neat and use radically different methods from simple counting arguments, so I hope you enjoy them. You may find that the problem of finding a succinct/effective description of a family of lower bound graphs is indeed interesting.
The best bounds I know of are due to Tom Bohman for $R(k,4)$ and Bohman and Peter Keevash for $R(k,5)$ and beyond. Both rely on using the differential equations method to analyze the following process: Start with the empty graph, and at each step add an edge uniformly at random among all edges which do not create a $K_t$. The bounds they achieve are
$$R(k,t) \geq c_t \left( \frac{k}{\log k} \right)^{\frac{t+1}{2}} (\log k)^{\frac{1}{t-2}}$$
The final term in this product corresponds to the improvement over the bounds obtained using the Local Lemma. For $t=3$ it matches Kim's bound up to a constant factor.
Best Answer
MathWorld has a pretty decent list (scroll down in the link) and cites numerous papers with good bounds
http://mathworld.wolfram.com/RamseyNumber.html