A Deputy Prime Minister is the deputy of a Prime Minister, and a member of a nation's cabinet. The position is often likened to that of a vice president, but is significantly different, though both positions are "number two" offices. The states of Australia and provinces of Canada each have the analogous office of Deputy Premier.
Deputy Prime Ministers traditionally fill in for the Prime Minister when he is absent or incapable of exercising his power. For this reason the Deputy PM is often asked to succeed to the Prime Minister's office following the PM's sudden death or unexpected resignation, although this is not necessarily constitutionally mandated.
Deputy Prime Minister is often a job that is held simultaneously with another ministry, and is usually given to one of the most senior, experienced ministers of the cabinet.