An "angel investor" is a term used for an investor in an early stage start up who, usually, takes some form of equity position within the business. An angel investor can be anyone who injects badly needed cash into a company when there are few other options. It is more of a term of art than, say, "Accredited Investor."
Typically, an angel investor will inject capital into a business to either help it get started or to help it over come some stage of growth. This person is usually a friend or family member but they can also be an interested outside investor who wants to see the business succeed. About the only requirement to be an angel investor is to have the money necessary to risk on such an early stage business and the desire to use it.