Welcome to the Apple Support Communities. I understand you're curious about this symbol next to your texts, and the good news is, it's pretty easy to deal with once you know what you're looking for. What you're seeing is a feature known as Do Not Disturb, which prevents notifications and alerts during times you might not want to receive them (such as during an event, at night when you're trying to sleep, or while you're driving). Do Not Disturb can be enabled as a general setting, which effects all notifications and alerts across the device; find out more about that and how to disable it here: Use Do Not Disturb on your iPhone, iPad, and iPod touch - Apple Support
Alternatively, Do Not Disturb can be enabled for individual message threads or contacts, which it sounds like is what you're seeing. Check this out:
Hide alerts
To stop receiving message notifications, you can use Hide Alerts. Go to the conversation that you want to mute, tap , then turn on Hide Alerts and tap Done. When Hide Alerts is on, will appear next to the conversation. This stops notifications only for that message conversation, not your device. You’ll still receive all other messages and see notifications for them on your Lock screen.
You can also hide alerts for all your conversations by turning on Do Not Disturb. Go to Settings and tap Do Not Disturb or Scheduled to set up a specific timeframe. This stops all notifications on your device.
Use Messages on your iPhone, iPad, or iPod touch - Apple Support
Either way, using these articles you should be able to get rid of the moon moving forward. Just let me know if you run into trouble while following these steps so I can keep working on it with you.
Cheers!