What is the relationship between patriarchy and sexism?
Patriarchy is regarded, by many feminist academics, as the dwelling-place and safeguard of modern sexism.
The concept of patriarchy far exceeds a short form answer. However it can be summed up simply in the idea of a masculinity-based hegemonic and overarching structure in society which prevades and instructs most, if not all, social interactions.
If patriarchy is assumed to exist, it yields a way of explaining sexism in modern society. That is, despite the generally agreed-upon stance that the sexes are equal, people continue to act in ways that indicate an assumed superiority of masculinity or at least an assumption of greater competence in men.
As an example: it is readily observed that women in similar positions to men in a workforce will often receive lower wages. This is explained in two ways: women ask for raises less often, and women are regarded as generally less worthy of raises.
This idea relates to the idea of patriarchy because as viewed by academia, patriarchy, that is to say the distinction between what it means to act like a man and a woman, is the underlying structure that informs the ideals of how a woman "should act" and thus teaches women to be less demanding in their careers. Further, patriarchy again defines the default successful employee as male. This predisposes managers to regarding female employees as less effective and thus less worthy of wage increases.