Text message and iMessage? What's the difference?

My mom gave me her old iPhone to have and AT&T disabled the phone when she got we new one but I put my SIM card in from my old phone and it works perfectly. I've been texting from it but in some conversations it says text message with green bubbles or iMessage with blue bubbles. I have unlimited texting so do I get charged to send or receive messages with either text or iMessage? And my dad says he has a detailed bill so he can read text messages I have nothing to hide but I was curious to see if that was a tulle true, and if I have wifi can I use the Internet on my phone without it charging to the data plan? Or does it cost even when I have wifi? Thanks!

Imessage is always free, even internationally. Even if you have unlimited texting those you text may not, so it helps it be free for them.

Text messages go through AT&T just as they always have.iMessage is Apple's messaging service, and is automatically used if you're messaging someone else who has an iPhone.iMessage uses your internet connection, not normal text message, and won't show up on the bill as a text message. Also, the bill will only show the phone numbers you're texting with, not the text of the message.
Yes, as long as your phone is connected to wifi your data usage isn't being billed to your data plan.

IMessaging uses the data plan to send messages between two devices, they both have to be iPhones, and both have to be either connected to a wi fi hot spot or have data credit.