What does half moon represent next to message?

https://www.google.com/search?q=how+to+turn+off+do+not+disturb+on+text+message&ie=UTF-8&oe=UTF-8&hl=en-us&client=safari#kpvalbx=_TzpbYZ_gCdeFwbkPyJiloAM12

sorry I should have posted this video. It shows you how to turn it off and on. Basically you go to the contact and to tap info, then you tap hide alerts. Turning this on or off with remove do not disturb on the individual. Have a great day!

It means you have enable do not disturb on the particular individual. It will mute notifications from this individual.