This can happen when a customer choses PayPal Website Payments Standard as the payment option.

There are one of two ways that your store won't be updated when a customer completes the order at Paypal.  The first is if the customer doesn't click on 'Continue' at the end of process on PayPal.  PayPal requires the customer to click on continue in order to send a response back to our system to complete the order in your store.  If the customer doesn't do that, your store doesn't get updated.

The second way is if Instant Payment Notification (IPN) isn't setup correctly on PayPal.  IPN connects back to your store and lets the system know that the payment went through.  If IPN isn't setup, then the transaction completes on PayPal, but the store is never updated and the order never completes.  To find out more about how to setup IPN, you should visit PayPal.  During the process of setting up IPN, you may be asked for a URL to link back to your store.  This URL should point to a response file that is on your store.  The URL looks something like this:

https://www#.corecommerce.com/~yourstore/orderSubmitted.php (the # in the www# would be replaced with the number of the server you are on).   If you are using your own SSL certficate it would be https://www.mystoredomain.com/orderSubmitted.php

You would change the domain name to match what your store uses.  Once you have saved that URL, PayPal will be able to update your store when a payment is made through their system.