A
person who has been
elected or otherwise
chosen as the new
president of an
organization or
state, but whose term in office has not yet begun.
In US government, the period during which the president-elect usually prepares for the transition is also the lame duck period of the outgoing President.
Some non-governmental organizations (wisely, IMO) make president-elect an official position, prescribing a term of a year in which the president brings the president-elect up to speed.