If you are experiencing a double sound alert for new text messages, calendar notifications, and other apps, then follow the instructions below to fix this problem.
Step 1) Hard reset your phone. To do this, simply press and hold the sleep/wake button and the home button at the same time until the Apple logo appears. This solution is very simple and fixes most instances of this issue.
Step 2) There may be a problem with iMessage on the new iOS 5. Go to Settings > Messages. Turn off iMessage. Reboot your phone (using the method above). Test to see if the problem is gone. Send a text to yourself and quickly press the home button to make sure you are not receiving the overlapping alert. Go Back into Settings > Messages. Turn iMessage back On. Send another text to yourself to clarify that the issue is gone.
Step 3) There might also be a problem with the way notifications are handled on the lock screen. Open Settings > Notifications > click on Messages > change "View in Lock Screen" to OFF. This will eliminate the double alert problem.
Step 4) If the problem still exists, you will have to restore your phone (guide here). However, do not choose to restore from a backup as the problem will still exist. Set up your iPhone as a new phone and then test out the notifications to see if they still double.
UPDATE: Reader, Kenneth Wu has pointed out that in Settings > Notifications > Messages > Repeat Alert, you can set the number of times that you would like to be notified of the same alert. Simply set this to Never if you would only like to hear your alert once. Thanks for pointing that out, Kenneth!