Currency convertibility means the freedom to convert domestic currency (rupee) into other international currencies (like Dollars etc.)
Before we discuss it further, read- What is current account and capital account?
Current account convertibility means freedom to convert rupee into dollars etc and vice versa for export and import of goods and services. It also includes freedom to convert currencies to make/ receive unilateral transfers like gifts, donations, etc and to pay/ receive interest, dividend, etc.
In India, there is full current account convertibility since August 20, 1993.
India had moved towards a market-determined exchange rate since March 1993. Then the RBI announced in August 1993 that, effective from August 20, India has become fully convertible on the current account. This was after India accepted the status and obligations of Article VIII with the IMF. It was a mere formality as India had already come very close to Current Account Convertibility.
Capital account convertibility is the freedom of foreign investors to purchase Indian financial assets (shares, bonds, etc.) and that of the Indian investors to purchase foreign financial assets. It involves the freedom to invest in financial assets.
There is partial capital account convertibility in India. It means there are certain restrictions on the movement of capital. Though India has been liberalising its capital accounts since the launch of economic reforms in 1991, it has adopted a cautious approach towards full convertibility, especially after the 1997 Asian financial crisis (which was exacerbated because the countries affected had full capital account convertibility) and the financial crisis of 2008 (which led to huge foreign capital outflows from emerging countries)