A company that distributes and redeems securities it issues. The most common open-end management companies are mutual fund companies which sell and redeem shares at the net asset value per share.
This is just a fancy legal name for a mutual fund. An investor in an open-end fund essentially pools his/her money with other investors in order to attain economies of scale, professional management, etc.

This differs from a closed-end fund which has a limited number of shares available. Unlike with open-end funds, an investor in a closed-end fund typically sells his/her shares on the open market to another investor instead of back to the fund company.