While public equity is a highly scrutinized and regulated dance, private equity is a secret dark corner of the dance floor, where nobody is supposed to look.
Private equity involves the sale of stock from a company to private investors who must sign "big boy letters," stating that if the company goes completely bankrupt, they are big boys and girls and they knew that there was massive risk when they invested in the first place. Why this kind of treatment vs. the very easy way Joe Sixpack can buy 100 shares of Coca Cola? Because too many fraudsters ruined the dance for too many excitable teenagers greedily seeking to make returns—and ending up broke.
Generally speaking, private equity comes in two forms:
The first form is called growth equity, generally attributed to fast-growing young technology companies simply seeking a late stage investment round, but who are not ready to be a fully public company... yet. Private equity growth rounds tend to be the last private rounds before the IPO.
The second form of private equity revolves around a combination of equity and debt, buying "fallen angels." That is, a company that was awesome 25 years ago and traded at 30x earnings became short-term greedy instead of long-term greedy and, over time, lost market share and mojo. In the process, their multiple declined to only 6x earnings. A private equity group works with a bank or banks to raise a large amount of debt, buying this company, taking it private...such that the private equity invested in it is highly leveraged. When the company goes public 3 or 4 years later, investors can make some 5-10-20x their initial investment if things go well. This latter form is a vastly different skill than growth equity investing, and is usually the primary attribution people consider when they hear the phrase private equity.