Functions are mapping from a preimage domain A to image codomain B. Each element in A must map to exactly one element in B.
What do you call a rule that doesn't map an element in A to exactly one element in B, but may map it to multiple elements in B? What would such a mapping or rule be called? "A relation that's not a function" sounds kind of clunky.
Best Answer
(1) It is perhaps worth noting that e.g. Hardy in his classic A Course of Pure Mathematics, talking of the case where $y$ is a function of $x$, writes that the principle "to each value of x ... corresponds one and only one value of y" is "by no means involved in the general idea of a function". Hardy, then, allows many-valued functions as genuine functions. Some modern writers on complex analysis still talk that way.
(2) Tim Gowers writes interestingly on what he calls multifunctions (i.e. many-valued functions) here, http://gowers.wordpress.com/2009/06/08/why-arent-all-functions-well-defined/
(3) There's nothing logically unmanageable about the idea of functions taking plural values -- see e.g. the treatment in Oliver and Smiley, Plural Logic.