Private equity simply means shares (of ownership) in a company that are not listed and traded in a stock market. Most businesses are privately held.
Private equity groups are investment groups that own private companies. They are formed as limited partnerships available to large investors and the organisers and managers typically get management fees and 20% of the profits (called a carried interest).
Depending on its purpose, a private equity firm may buy and manage private companies, it may be a venture capital group that invests in new business ventures, it may be a growth capital fund that invests in small private companies that are past the start up stage and need investment to expand without going public or before going public. Private equity groups may actually acquire publicly traded companies and take them private if they think they are undervalued by the stock market in what are usually leveraged buyouts where most of the money used to buy the company is borrowed.
An individual private equity firm will have a stated business strategy and objectives that they say they will follow to bring superior returns to their investors. For example, they may focus on companies with good positions in growing niche markets. Or they may specialise in a particular industry or sector like health care or information technology where they believe their knowledge will give them an advantage in finding or managing companies in that field.
In other words, when people speak of private equity groups, they usually are speaking of a firm that buys private companies outright or buys public companies, although, strictly speaking a venture capital fund or a growth capital fund is also a private equity fund. The key is that the equity is not shares that are publicly traded in a stock market.
And as I said above private equity funds are actually limited partnerships with the managers as the general partners and the investors being the limited partners. (Limited meaning their personal liability is limited to losing whatever they invest.) The investors in private equity funds are institutions and high net worth individuals.
To give you a simplified example, let's say you and some partners start a private equity fund. You raise maybe $200 million from investors ranging from rich people to pension funds and with it you go and buy (usually fairly small) privately owned businesses with the money you raised plus probably borrowed money. You assemble a portfolio of business and you and your partners will earn a percentage of the profits from those businesses, usually 20%, with the rest going to the limited partners. The limited partners invested with you because you persuaded them you can earn higher returns (adjusting for risk) than they would make in the stock market.