A casting vote is a vote that someone may exercise to resolve a deadlock. A casting vote is typically by the presiding officer of a council, legislative body, committee, etc., and may only be exercised to break a deadlock.
Examples of officers who hold casting votes are the Speaker of the British House of Commons and the President of the United States Senate (an ex-officio role of the Vice President of the United States).
In some legislatures, a casting vote may be exercised however the presiding officer wishes. An example is the Vice President of the United States, who may exercise his casting vote in the Senate according to his party affiliation or according to his own personal beliefs; by virtue of the Vice President's casting vote, when the Senate as elected is equally divided between two parties, the Vice President's party is able to serve as the official majority party in the Senate.
In some other legislatures, by contrast, a casting vote can only be exercised according to strict rules or conventions. For example, the Speaker of the British House of Commons (and analogous positions in most Westminster Parliaments) is expected by constitutional convention to follow Speaker Denison's rule, i.e. to vote to allow further discussion, if this is possible, and otherwise to vote in favour of the status quo.
In the United States, the concept of a casting vote is not used in Robert's Rules of Order.