Business is the activity of making one's living or making money by producing or buying and selling goods or services. Simply put, it is "any activity or enterprise entered into for profit. It does not mean it is a company, a corporation, partnership, or have any such formal organization, but it can range from a street peddler to General Motors." The term is also often used colloquially (but not by lawyers or public officials) to refer to a company, but this article will not deal with that sense of the word.
Management (or managing) is the administration of an organization, whether it is a business, a not-for-profit organization, or government body. Management includes the activities of setting the strategy of an organization and coordinating the efforts of its employees (or of volunteers) to accomplish its objectives through the application of available resources, such as financial, natural, technological, and human resources. The term "management" may also refer to those people who manage an organization.
You manage things, you lead people. We went overboard on management and forgot about leadership. It might help if we ran the MBAs out of Washington.
Grace Hopper (1987), quoted in "The Wit and Wisdom of Grace Hopper" by Philip Schieber, OCLC Newsletter, No. 167 (March/April 1987)