Private Equity (PE) as it sounds is basically a way of raising equity privately. Private Equity is a general term used to describe all kinds of a fund that pools money from a bunch of investors that are used to acquire stakes in companies.
Technically, Venture Capital is Private Equity. But PE is often associated with the funds trolling for mature, revenue-generating companies in need of some revitalization in order to become worth much more. While Venture Capital often goes in younger companies involved in unproven, cutting-edge technologies, funds described as Private Equity are more attracted to established businesses.