What does it mean when I send someone a message and it doesn’t say delivered. But I get notified at the bottom of the page that they are on do not disturb mode.

It means they are in "Do not disturb" mode and will not get notifications of new messages, so it may be a while before they answer.

They may be doing something they need to focus on, and don't want to be distracted by messages or calls, so they turn on "do not disturb mode".

Link->Use Do Not Disturb on your iPhone, iPad, and iPod touch - Apple Support