We can talk a lot about why we like Discourse so much as a modern day forum (and we do), but seeing is believing. So have a look at 5 succesful examples of Discourse forums ...
Of course Discourse itself uses Discourse to discuss the product, communicate new features, function as a helpdesk, etcetera. A good example of eating your own dogfood.
Music lovers like to discuss a lot. That's more then visible at Drowned In Sound. Have a look at the community. And see how nice it is that embedding music and video is so easy in Discourse. It's only a matter of dropping a link.
Discourse works really good in an education environment. Because teachers and pupils like to talk about lessons, like to share knowledge. Code Academy is a great example.
Good companies want a strong connection with their customers. The Bank of New Zealand does. It uses Discourse to answer questions, to ask about whishes and experiences, etcetera.
This is a great example: pilots answer all the questions people have about flying. Feed the passion, share the knowledge.