The green icon on your phone w/the white "caption" icon is where you are sending text messages to and from.
When you send receive messages w/someone who has an iphone note that color of their messages is BLUE. That means your phone is sending and receiving via iMessage - unique to Apple and can be sent / received w/o any cell service.
Some of your messages have a green background. These are going to or from people who do NOT have an iphone. You need cell service to receive these. They are being sent via SMS. If you are in Europe and turn off your cell service you wouldn't be able to message them or receive messages from them.