A finite state transducer is a machine model, much like a finite state automaton, but equipped with two tapes, one for input and one for output. Each specific finite state automaton defines a binary relation between input and output strings.
Your assumption is right: for each input string we look at computations that match that string and output the corresponding strings on the second tape. This process can be highly nondeterministic, as symbols can be deleted and inserted at will (when the automaton is programmed to do so).
This definition is extended to languages: each transducers defines a binary relation on languages too.
Transducers can be found for a very broad class of tasks: homomorphisms, inverse homomorphisms, intersection or quotient with (a fixed) regular language.
Finite state transducers are nice devices, e.g., they can be programmed to deleted every second $a$ in a string, but only for strings ending in a $b$.
The wikipedia article on FST ignores a very important basic fact on transductions. They happen to be equal to the class of operations built from homomorphisms, inverse homomorphisms and intersection with regular languages. Each family closed under these operations is called a trio, and such a class is then closed under all FST's. Examples are the family of regular languages and the family of context-free languages. So as a consequence, given a CFL, the language we obtain by erasing every second $a$ (for strings that end in a $b$) is again context-free.