Your iPhone has multiple messaging options to give your network ways to contact you. Your phone callers are given the option to save recorded messages for you to retrieve later. These messages are called voicemails and are stored in your "Phone" application. Once you retrieve your voicemail, you can delete it to save storage space. Simply selecting the voicemail and pressing the delete button gets the voicemail out of your voicemail list.
Open the "Phone" application on your iPhone by tapping on the "Phone" app icon on the home screen.
Click the "Voicemail" icon at the bottom right of the "Phone" application.
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Press the voicemail that you want to delete.
Click the "Delete" button.