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.
Politically, the Indian subcontinent includes Bangladesh, Bhutan, India, Maldives, Nepal, Pakistan and Sri Lanka. Sometimes, the geographical term 'Indian subcontinent' is used interchangeably with 'South Asia', although that last term is used typically as a political term and is also used to include Afghanistan.