The following information is from our handy-dandy help file titled: Your Settings.
The Notification Options page allows you to configure how you should be notified of certain events, such as a reply to a topic, a private message, or a new topic. You can configure how you would like to recieve these notifications as well, such as via the mobile application, email, or inline notifications.
If you're logged on to the forum in your iPhone's browser, at the top of the page in the blue bar is a set of four squares. If you click those bars while logged in to the forum, you'll see a section called Notifications. You can see all your notifications there.
I'm not sure about the push notifications and how they would appear any differently. At one point, IPS had an actual iPhone app, but that is not currently available. Push notifications may reference the unavailable app, instead of referencing the mobile skin (which is what people get now when they come to the forum through their phone's browser).