I've heard lay definitions of "patriarchy" as broad as "any exploitation of any group by any other group or individual" and as narrow as "oppression of women by men." What I mean when I say the term is pretty much what Merriam-Webster says it means: "a social organization marked by the supremacy of the father in the clan or family, the legal dependence of wives and children, and the reckoning of descent and inheritance in the male line, or broadly, control by men of a disproportionately large share of power; or, a society or institution organized according to the principles or practices of patriarchy."
So: Patriarchy is a social organization that assumes men take the role of caretakers, custodians, providers and rulers, and that women and children take the roles of dependents.
Unfortunately, this term carries the connotation that all men are more advantaged than all women. Women might be more advantaged than men in some situations; women of higher social classes might be more advantaged than men of lower classes; women and men of advantaged or disadvantaged races and of varying social ranks might have quite complicated dominance hierarchies. So, the term "patriarchy" seems a little too simplistic.
Maybe "kyriarchy," meaning "a complex pyramidal system of intersecting multiplicative social structures of superordination and subordination, of ruling and oppression," would be better?
This term is ungendered, removing the common misinterpretation that it's men who are oppressing women rather than men and women who are engaged in complex social interactions dependent on gender assumptions of each; and it takes into account the complexities of society and does not assume that one gender will always have the advantage of the other (or even that gender is the most indicative factor in determining advantage or disadvantage).
Seems pretty awesome to me.