You don't need the application to be running in order to receive SMS messages. As soon as you get it, the iPhone pops up with an alert (and sound) that tells you that you received an SMS message. It also updates the number next to the SMS application icon.
Another example is: AIM
If AIM had push, you could log into AIM through the application, then press the home button and do other stuff. When you receive a message, an alert, sound, and badge (white number in red circle next the application) will pop up telling you that you received a message (most likely it will show a small portion of the message and the person that sent it).
Currently, you can only receive AIM messages if you are running the AIM application. AIM, Palringo, and MobileChat have kind of made up for this by keeping you logged on their server even after you quit the program. They then store all received messages until you open the program again. Once you do, you receive all the messages that were sent while you weren't in the application.
Currently - When it is running, it can receive instant messages. When it is closed, nothing gets to the phone until you open the app again.
With Push - Someone sends you an IM while AIM is closed. Just like with text-messages, you'd get a sound alert and a red dot on the AIM application. You can open it and get your IM, even though the app was closed when it was sent.
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So for that example, AIM could be made to work just like e-mail or the text-messge application. Currently, it is not like either of those.