If a message is stuck in the Outbox, please go to File and choose Work Offline. Close outlook and reopen it. Now you can delete the email. This happens usually to emails with attachments; it may be time to install a new Outlook.
If Outlook is not downloading all of your messages, and the quota is good, then you may have other mail programs (like Blackberry) downloading the emails and thus removing the copies.
If Outlook is downloading multiple copies of your messages, then most likely your Outlook program is broken. I suggest downloading a new copy to install, or there may be a patch provided by Microsoft.
If Outlook is trying to download a message, but gets stuck, then the message is likely too large or has a large attachment. I suggest you delete the problematic message from webmail and try again.
Help pages: Mail setup for Outlook and Thunderbird, and troubleshooting guide