What can be solved with semidefinite programming that can't be solved with linear programming? I'm familiar with linear programs in that they can solve problems with linear objective functions and linear constraints. But what can semidefinite programming solve that linear programming can't? I already know that semidefinite programs are a generalization of linear programs. 

Also, how does one recognize a problem that can be solved using semidefinite programming? What is a typical problem that semidefinite programming is used for that couldn't be solved via linear programming? 

Thanks very much for any response.
A typical problem is MaxCut: output a cut in a graph that (approximately) maximizes the number of edges cut. Goemans and Williamson showed an SDP approximates the value of MaxCut to within a factor at least 0.878. Recently, Chan, Lee, Raghavendra, and Steurer showed that for a natural linear encoding of the MaxCut problem, all polynomial size LPs achieve approximation no better than 0.5. 

It's hard to say concisely what kind of problems usually benefit from an SDP. A systematic approach to constructing SDP relaxations is through hierarchies, the most powerful of which is the Lasserre hierarchy: see Rothvoß's survey for a nice introduction. By now there are too many examples of successes of SDPs in optimization to list. Also, Raghavendra showed that one particular SDP gives the best approximation to all MaxCSP problems, if the Unique Games conjecture is true. 

Check the books of Gaertner and Matousek, chapters 6 and 13 of Willimson and Shmoys' book, Lovasz's survey.