Terms with Default Definitions found in this post
A Patriarchal Culture, or Patriarchy is a culture in which Men are the Privileged Gender Class. Specifically, the culture is Srolian, Govian, Secoian, and Agentian. The definition itself was discussed in a series of posts, and summarized here. See Privilege, Oppression.
Feminism is a collection of movements and ideologies aimed at defining, establishing, and defending political, economic, and social rights for Women.
A Feminist is someone who identifies as a Feminist, believes that social inequality exists against Women, and supports movements aimed at defining, establishing, and defending political, economic, and social rights for Women.
Privilege is social inequality that is advantageous to members of a particular Class, possibly to the detriment of other Class. A Class is said to be Privileged if members of the Class have a net advantage in gaining and maintaining social power, and material resources, than does another Class of the same Intersectional Axis. People within a Privileged Class are said to have Privilege. If you are told to "Check your privilege", you are being told to recognize that you are Privileged, and do not experience Oppression, and therefore your recent remarks have been ill received.
The Glossary of Default Definitions can be found here