Well, that Wiki entry is a little overly broad and confusing, and since i worked in the space for over a decade on several sides of it, here is an explanation from 10,000 feet.
Private equity is a term used to describe funds that were raised through private offerings to qualified investors only ($1 million in net worth, annual income $200k) with a long term commitment on those funds of up to 10 years+ with investors having no control on when and how the funds are disbursed or invested other than is specified in the offering memorandum.
Historical uses of private equity funds has been Venture Capital and Leveraged Buyouts Funds, which are very different in their strategy and tactics. There are others, but i'll focus on the two which make up most of the industry.
VC's make their money on earlier stage high growth companies in the hope of taking them public in an IPO or sold to a strategic buyer / corporation. They fund many different stages of growth and rounds of investments and security classes, and at the end of the day live and die by ultimate proof of concept (and growth) of the underlying companies. There is little if any debt taken on in these deals.
Leveraged Buyouts make money buying proven, mature and high cash flow producing companies. The deal is worked to where a bank will come up with (e.g.) 70% of the purchase price and the fund will put in 30% cash equity and control 100% of the company. The company makes a return on its capital by 1) paying its debt down, and 2) hiring a star manager to improve sales and earnings. Success in company growth could also have the potential to see a 3) purchase multiple increase on the valuation of the company at sale, - this scenario of succeeding in all three categories would be considered a "home run".
Also, don't forget in an LBO world, the fund / partners can take hefty annual management fee from the company, as well as any manner of advisory, transaction or consulting fees it sees fit.
Finally, it makes the big money by an 80/20 split of all profits above an preferred aka hurdle rate (the first 10% of profits).
The VC model needs only 3 out of 10 investments to succeeed, and one or two earning tremendous windfall profits in excess of the other 7 that fail to return any.
Contrary, the LBO model usually needs at least 7 out of 10 to succeed (as far as generalities go) as they are a lower risk higher success rate investing model.