For the most complete documentation for upgrading an existing Drupal site, see http://drupal.org/upgrade.
Instructions on how to update Drupal core with a newer version of Drupal of the same category or same base version [ ie Drupal 6/7/8]
- Make a backup of your Drupal instance. (For ex: with MySQL)
- Download the latest release of your current Drupal version.
- Extract the [tar ball or zip] Drupal package.
- Set your site on maintenance mode (For ex: How on D7)
- Delete all the files & folders inside your original Drupal instance except for /sites folder and any custom files you added elsewhere.
- Copy all the folders and files except /sites from inside the extracted Drupal package [tar ball or zip package] into your original Drupal instance.
- If the update release includes changes to settings.php replace old settings.php in .../sites/default/ with the new one, and edit site-specific entries (eg database name, user, and password)
- If you modified files such as .htaccess or robots.txt re-apply those changes to the new files.
- Login to your site as administrator or as user no 1
- Run update.php by navigating to http://...yourdrupalsitename/update.php
- Follow the process to update your Drupal instance
- Disable maintenance mode