A very rough way to distinguish ground states from metastable states (in the context of Spin glasses, as an example) is as follows:

escaping a metastable state requires overturning a fixed number of spins, independent of the system size.

On the other hand, escaping a ground state requires overturning a number of spins that grows with the system size.

Metastable states are surrounded by relatively low “energy barriers,” of $O(1)$ energy, while ground states, or true global minima, are surrounded by energy barriers that grow with the size of the system.

This phenomenon (and its consequences) was dubbed “broken ergodicity” by Palmer [Broken ergodicity in spin glasses]