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'Convertibility' is the quality of money which is officially backed by government reserves of a precious metal, usually by the gold standard. Under convertibility, currency is seen as more reliable and less prone to exchange-rate fluctuations (though gold, of course, also changes in value). The Bretton Woods Institutions were set up partially to allow countries to peg their currencies to the US dollar instead of their own gold reserves; the U.S. eventually abandoned the gold standard, and thus convertibility, in 1971.
It can also refer to the convertibility of one currency into another. Some countries pass laws restricting the legal exchange rates or requiring permits to exchange more than a certain amount. Thus, those countries' currencies are not fully convertible.