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A place is classed as a subcontinent when it is part of a continent but is considered either geographically or politically as an independent entity or it is smaller than a continent.

India is often called a subcontinent because it is a distinct landmass, not just a country. While it has many features of a continent, it is not as big as one, so is not considered a continent.
India was once a continent (or at least a very large island). Due to continental drift, India moved north and became part of Asia.
India is now joined with the continent of Asia, but is a distinct area separated by the Himalayas. Originally it was a separate geologic plate, but has collided and merged with Asia.