In the British Government, it is the office of the most Senior finance minister in the Prime Ministers cabinet. The Exchequer is charged with the management of the public income (read: Taxes),the expenditure of those monies and the general responsibility of Her Majesty's Treasury.
Below the Exchequer are the Chief Secretary of the Treasury, Paymaster General, Financial Secretary and Economic Secretary.
A good comparison for those familiar with the United States government would the the office of the Secretary of the Treasury or the General Accounting Office.
As of September 2001, the Exchequer is the Honorable Gordon Brown who was appointed as to the office on 2 May 1997. As per tradition, he resides at Number 11, Downing Street.
Sources: www.hm-treasury.gov.uk, Merriam-Webster new world encyclopaedia, 2000 edition.