A big classification result that I'm aware of is due to Gromov and Lawson.
Theorem. Let $M$ be a compact simply connected manifold of dimension $\geq 5$, which is not a spin. Then $M$ admits a metric of positive scalar curvature. A spin manifold of dimension $\geq 5$ carries a metric of positive scalar curvature iff its $\alpha$-genus is zero.
The Seiberg-Witten invariants provide special obstructions to existence of a metric of positive scalar curvature in dimension 4.
There are two good survey articles on the subject by J. Rosenberg (link 1) and S. Stolz (link 2).