Convertibility is the extent to which a currency can be freely exchanged for others. It is usually described in two halves: current account convertibility, meaning residents may buy foreign exchange for trade and similar ordinary transactions, and capital account convertibility, meaning money may move in and out freely for investment. A currency can be fully convertible on the first and heavily restricted on the second.
Convertibility is what decides whether a currency has a deep market at all. Fully convertible currencies are held and traded worldwide, quoted continuously and available in size; partially convertible ones are traded mainly by parties with a domestic reason to hold them, and are often reachable from outside only through instruments that settle in another currency. The classification is a matter of a country's own law and regulations rather than an assessment anyone makes from market behaviour, and it can be widened or narrowed by the authorities that set it.
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How to Check the Way a Currency Is Managed
The regime behind a currency is published, not inferred. The four primary sources that say how a rate is governed, what each one can and cannot tell you, and how to keep the answer from going stale.