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'Metastability' is the ability of a non-equilibrium state to persist for some period of time.
'Metastability in molecules' is the ability of a non-equilibrium chemical state to persist for a long period of time.
Usually metastability is due to a relatively slow phase transformation. For example at room temperature diamonds are metastable because the phase transformation to the stable graphite form is extremely slow. At higher temperatures the rate of phase transformation is increased and the diamond will transform to graphite.
Martensite is a metastable phase used to control the hardness of most steel. The bonds between the building blocks of polymers such as DNA, RNA and proteins are also metastable.
IUPAC recommend that the term "metastable" be avoided and "transient" be used instead because "metastable" can misleadingly associate a thermodynamic term to a kinetic property, even though most transients are thermodynamically unstable with respect to reactants or products^[1].

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1. http://goldbook.iupac.org/T06451.html