One of the core ideas in YALMIP is to rely on external solvers for the low-level numerical solution of optimization problem. YALMIP concentrates on efficient modeling and high-level algorithms.
A simple categorization is as follows (the definitions of free and commercial depends slightly on the solver, please see the specific comments in the solver description)
Linear programming (free)
Mixed Integer Linear programming (free)
Linear programming (commercial)
Mixed Integer Linear programming (commercial)
Quadratic programming (free)
Quadratic programming (commercial)
Mixed Integer Quadratic programming (commercial)
Second-order cone programming (free)
Second-order cone programming (commercial)
Mixed Integer Second-order cone programming (commercial)
Semidefinite programming (free)
Semidefinite programming (commercial)
General nonlinear programming and other solvers
Linear programming can be solved by quadratic programming which can be solved by second-order cone programming which can be solved by semidefinite programming. Hence, in theory, you only need a semidefinite programming solver if you only solve linear problems.
A recommended installation if you mainly intend to solve SDPs and perhaps some small to medium sized LPs and QPs is SeDuMi or SDPT3 (these solvers are by no means the best solvers, but they support second order cones which makes them fairly efficient also for QPs).
If you are in academia, MOSEK is convenient since it gives you a very competetive MILP, MIQP, MISOCP and SDP solver in one package.
If you intend to solve large problems or other problem classes, you are advised to download several solvers to find one that works best for your problem class.
And finally, there are no free lunches and you get what you pay for, most often...
By exploiting the optimization infrastructure in YALMIP, it is fairly easy to develop algorithms based on the external solvers. This has motivated development of mixed integer conic solvers (bnb, cutsdp), global solvers for general nonlinear problems (bmibnb), indefinite quadratic programming (bmibnb, kktqp ), sum-of-squares modules (solvesos), just to mention some of the most important contributions in YALMIP.