Alert messages are saved in the Notification Center on the iPhone. You can view the Notification Center at any time by swiping down from the top of the screen. All recent alerts and notifications are displayed and can be customized. For more information on customizing alerts and the Notification Center, click this link to see my detailed article about customizing the iPhone notifications.
You can exit the Notification Center by swiping up from the bottom of the screen or by pressing the Home button.