Do I need to upgrade my database?
After uploading the new files, visit http://example.com/update.php, read and act upon the instructions, then click through to "run the database upgrade script". You may expand the "Select version" fieldset to review the updates Drupal automatically selected or simply click update.
Drupal 5 and later will also alert you to run update.php if it is detected that your database is no longer up to date. The alert will appear on the Administration page (admin) and the status report (admin/logs/status).
Do not Overwrite Anything Yet
If you're following this guide step by step, don't be mislead by the line "After uploading the new files..." etc. Don't upload/overwrite anything yet. Notice that in this case, making a backup is not just a healthy practice, its actually necessary as part of the upgrade. Its not optional. You -will- need some files from your backup later on (settings.php amongst them).