It occurs to me that as the U.S. political debate is getting deeper into the weeds of what Bain Capital in particular and the private equity indusrty in general do, that a lot of people are probably not that familiar with the term "private equity." But here's the deal, roughly speaking:
Most large businesses are "public," which is to say that at some point they registered with the Securities and Exchange Commission (SEC), did an Initial Public Offering (IPO), and their shares are bought and sold on stock exchanges. Lots of investment funds are dedicated in whole or in part to buying and selling shares in this way. Ownership of public companies is generally pretty distinct from management. The managers may or may not own shares in the firm, but the bulk of the shareholders/owners have nothing in particular to do with running the company. People buy and sell shares according to whether or not they think that's a smart financial decision, and managers go run the company subject to the very imperfect oversight of a Board of Directors.
A private equity fund is a vehicle for getting a bunch of money together to buy existing public companies and "take them private," typically (though not necessarily) with the goal of re-selling the firm later. There are a bunch of things that characteristically happen after the company is taken private, but the defining feature of the genre is simply that you're owning private companies rather than owning shares of a publicly traded company.