I would check the Notifications pref pane to check if it's allowed to post notifications.
Then the Security & Privacy preferences pane to ensure it has permissions to post things there.
Finally check the Mail , Calender etc etc preferences pane to make sure you have the twitter username and password correct (not sure what this pref pane's exact name is)
If you're having issues logging into twitter make sure www.twitter.com actually resolves to the twitter site in a web browser. If the site itself is down you won't be able to log in.