Connect with Discord
The checkbox "Remember me" allows you to stay connected to the website, even after a prologated period of inactivity or after the web browser has been quit. This permits you to avoid going back through the "Connection with Discord" square. Not ticking the checkbox will have no effect and you will be automatically disconnected.
Explicitly disconnecting from the website will disconnect you entirely. You will then be able to use another account or do something else.