It means that notifications of a message or event are initiated externally, causing the phone to alert you without any action required on your part. This is as opposed to you having to initiate a check for an event from your phone. "Push" comes from the notification being forced to the phone from the server. For example, if you go out and check for e-mail manually, or have the phone automatically check for e-mails at specified intervals, that's pull technology. But when you receive a text message alert without having done anything from your end, that's push. It can apply to many types of alerts and notifications.