Typically a Mutual Fund is an investment fund aimed at individual investors sponsored by an investment (or "mutual fund") house like Fidelity, Vanguard or T. Rowe Price. Each fund holds a "market basket" of stocks or bonds and individual investors buy into the fund by buying a share at "Net Asset Value," which is the total worth of the fund's holdings, calculated every day, divided by the number of shares outstanding. In other words, a mutual fund whose portfolio (value of all holdings) is worth a million dollars that has a hundred thousand shares outstanding will value those shares at ten dollars apiece. A typical stock-based mutual fund can earn its investors money in three ways: the dividends and capital gains that stocks pay out, and possible appreciation of the fund value per share.
For an individual investor, the advantage of owning a mutual fund is that s/he achieves diversity -- mutual funds own more than fifty stocks, on average -- that could not be achieved by buying a typical hundred shares of stock in only a few corporations. The disadvantages of such funds are that the "load" (sales commission) involved in buying or selling such funds can be considerable, and all funds incur some sort of service fees; that's how the investment house earns its money. Also, no "equity" or stock-based investment is guaranteed.
"My broker wants me to buy shares in something called an "open-end fund" but I don't know what that means."
"That's just a way to describe the majority of mutual funds, which remain open to all new investors who have the money to invest in them."