I am reading about water in the Lunar mantle, however it mentions fugacity a lot. What does it mean in this context?
'Fugacity' is effectively the equivalent of chemical activity, but for gases. Fugacity relates to the partial pressure of a particular volatile component of a system in the same way that chemical activity relates to the concentration of a given chemical phase.
Most often in geology, this will refer to 'fugacity of oxygen (fO2)' in a system - this is effectively a measure of how reactive oxygen is, and thus how oxidized a given system will be, usually with respect to the minerals that are stable in them.
At temperatures that most mantle processes operate on, aqueous fluid phases generally exist in a supercritical state, which we also apply the concept of fugacity to. So, if the text you are reading states something about 'fugacity of H2O', then it is likely referring to the reactivity of supercritical water vapor in the lunar mantle system.