What does i mean when there is a shaded moon icon next to certain contacts when they text

This means you have manually enabled Do Not Disturb for these recipients. As long as the feature is enabled, you will not be notified of new messages in these threads. The feature can be controlled from within the message by tapping the "More Info" icon in the top-right corner and toggling the Do Not Disturb option.

It means you have set that conversation on Do Not Disturb.

Open the conversation. Tap on the little "i" in the upper right corner. Turn Do Not Disturb to off.

Setting Do Not Disturb on a contact means you will receive messages from that person but you will not get notified of these messages.