You may want to see the little preview of text messages that turn up on your home screen when a new text (SMS) message comes in. To turn this on, head to your iPhone settings and go to:
General -> Passcode Lock -> Show SMS Preview
Turning on 'Show SMS Preview' will enable the text message preview on your iPhone. You will get an alert that the message came in, including the sender's name/number, as well as a portion of the message.