This is the place itself, the name that everything else hangs on. The source puts it in the mountains of north-east India, in West Bengal, and says it is the main town of the district around it. It is a proper noun, so it always takes a capital D and normally takes no article: you go to Darjeeling, you live in Darjeeling. Because the hills nearby grow the tea, the same word ends up on a tea packet, which is why a learner may meet the name long before they meet the place.

This is a tea named after the place it comes from, the way many foods and drinks are. The source calls it delicate, which is a word about taste: light rather than strong. Because it is a short form of Darjeeling tea, you can use either, and a menu may print the long name while a customer says the short one. It keeps its capital letter, since the name is still the name of a town. You will normally treat it as uncountable, some Darjeeling, a pot of Darjeeling, though a shop comparing kinds may talk about different Darjeelings.