The iPhone 6S and 6S Plus are shipped with a new feature that allows users to mute notifications pertaining to a specific user or conversation. Here are the instructions on how to enable the Do Not Disturb feature in the Messages app.
Silence Notifications for Specific Message Threads
Open the Messages app, and select the conversation you would like to mute.
Tap on the Details menu displayed at the top-right corner of your screen and toggle the Do Not Disturb switch to On.
Moving forward, your iPhone will no longer alert you when you receive new messages from the selected contact or message thread.