When you receive a new e-mail message on your iPhone or iPad, if your device is currently locked then the message will show up in Notification Center and display on your home screen. If you want to access your message directly, then instead of unlocking your device and going to Mail, you have several options available to you right on the home screen.
First, if you want to read your message, then as with any notice you can tap and slide it to the right on your home screen. Following this, you will be prompted to unlock your device, and then Mail will open and show you the message in full.
A quicker alternative to this is to do these actions right on the home screen. Instead of sliding to the right, slide your message to the left, and you will have three options: Mark as Read, Trash, and an “x” button for dismissing the notice to remove it from your list of notifications.
These options are also available for iMessages, where by sliding to the right on a received message and then unlocking your device, you will open the Messages program to continue your conversation; however, by sliding to the left, you will see the option to dismiss the notice, along with a Reply option where you can give a quick reply to the sender without needing to unlock your device.