I call the problem as DOM. Given a graph $G$ and an integer $d$, Decision problem DOM - "Does there exists a dominating set of size less than or equal to 'd' in G? " DOM is NP Complete problem. But, if we specify some class of graphs $G'$ and if we ask the same question, then DOM might be solved in polynomial time. (Eg : Given $C_n$, and an integer $d$ , you can verify DOM in polynomial time ) Till date , for which all classes of graphs, DOM problem can be solved in polynomial time ? [Let me know where I can find research papers worked on this topic]
I guess that related Wikipedia article is a rather complete and up to date source. According to it, a minimum dominating set can be found in linear time in series-parallel graphs. On the other hand, even for planar graphs the problem remains NP-hard, but it admits a polynomial-time approximation scheme (PTAS) and a fixed-parameter algorithm is known.