A one-to-one correspondence is another name for a bijection.
In economics, a correspondence between the sets A and B is usually thought of as a map f:A→P(B)from the elements of a set A into the set of all subsets of a set B (the power set of B). This is equivalent to the first definition. However, there is usually an additional property that for all a in A, f(a) is not empty. In other words, each element in A maps to a non-empty subset of B; or in terms of a relation R as subset of AxB, R projects to A surjectively.
An example of a correspondence in economics is the best response correspondence in game theory, which gives the optimal action for a player as a function of the strategies of all other players. If there is always a unique best action given what the other players are doing, then this is a function. If for some opponent's strategy, there is a set of best responses that are equally good, then this is a correspondence.
In algebraic geometry a correspondence between algebraic varieties V and W is in the same fashion a subset R of VxW, which is in addition required to be closed in the Zariski topology. It therefore means any relation that is defined by algebraic equations. There are some important examples, even when V and W are algebraic curves: for example the Hecke operators of modular form theory may be considered as correspondences of modular curves.