iMessage uses internet data, whether it is on wifi or cellular. It does not use very much data but it does use some. if your messages go to an iPhone it will fall under that. A non-iMessage device will still send SMS or MMS messages which may have a cost per message depending on your plan.
With an iPhone if your bubble is blue it's na iMessage, and if it's green it is a text message. Should it switch there will be a notification such as "SMS from" above the message.