If $G$ has a normal $p$-subgroup containing its centraliser then it only has one $p$-block. This goes back to Brauer.

If $G$ is simple and $p$ is odd, then $G$ has more than one $p$-block (Brockhaus and Michler, J. Algebra 1985). For $p=2$, I think the only examples with just one $2$-block are the sporadic groups $M_{22}$ and $M_{24}$, but I could be wrong.