Why can I only log in with firefox?
Login with Firefox.
While logged in on Firefox, open a separate window to www.example.com/user in Internet Explorer and try to log in.
If it fails, got back to the logged in session on Firefox and navigate to the Drupal Watchdog. See if any error appears.
Locate your PHP error logs, and see if they logged in any errors.
Edit your question and post any errors that you see.
...
Profit!


I feel this is a somewhat ambiguous error that could be being caused by almost anything, and there will be no way that anyone here is going to be able to psychically log in to to your server and figure out what exactly is causing that, without a more descriptive question. Checking what are your current watchdog/PHP errors is probably the best thing that you can do at the moment.