Common meaning:

Special rights given to a person or group.

Hidden meaning of the Equity religion: 

The belief that there are structural advantages given to members of dominant groups over members of oppressed ones. 

Discussion:

The Equityists believe that members of dominant groups receive structural advantages over members of oppressed groups. The evidence to prove that this must be happening is seen any time there are proportionally more members of the dominant group in a position than members of the marginalized one.

The reality is there may be many reasons that a dominant group has proportionally more members in a position than a marginalized one. Most of these reasons will have nothing to do with discrimination and bias.

The Equityists think that privilege is something everyone in the dominant group has and something no one in the oppressed group has. Equityist people accept, by faith, that privilege is happening and no evidence, other than unequal group levels, is necessary to prove it.

How exactly white heterosexual males living in the mountains of Appalachia partake in this mysterious privilege is never explained.

Claiming privilege is a good way to hide your jealousy.