Hello new here, What does the crescent moon icon next to the contact name mean? There is a moon icon similar to the do not disturb icon next to a contact name in my phone. What does that mean?

Do Not Disturb is on. To remove it, follow this path: Select the Message>Select Details>Toggle "Off" Do Not Disturb

That's usually do not disturb for the contact. There should be edit contact at the bottom and I. There you can remove the disturb