The equity of the publicly traded companies on the stock exchanges is called the public equity or common shares which are available to everyone. The equity part of the privately (not traded on stock exchanges) held companies is called the private equity. These equities are not easily available in the market and the investors can’t buy or sell whenever they want. Investors has to buy the equity part in the privately held companies from the owners as private equity and has to sell to other investors through the proper process.
As per the laymen terms, Private equity is money invested in companies that are not publicly traded on a stock exchange or invested as part of buyouts of publicly traded companies in order to make them private companies. Private equity investment always requires long term commitment as the liquidity is very low and the investors have to stay invested for some minimum period specified at the time of buying the private shares.
Private Equity Fund is the fund which invests in the privately held companies as private equity to be a part of their long term growth story. They always fix a minimum time frame for the investment at the time of acquiring the stake with the intention to sell the stake only after the minimum period at a very high profit.