Text Messages are muted
What does it mean when text messages are muted?
iPhone 6s Plus, iOS 7
Is this what you mean?
Keep it quiet. Tap Details, then turn on Do Not Disturb to mute notifications for the conversation.
http://help.apple.com/iphone/9/#/iph01ac18d71
It means you will not receive notification for that conversation (the messages will still arrive, however).