First of all you should be aware that you can use puTTY and drush to upgrade and update your site.
Minor update procedure
With a minor release update; such as from Drupal 7.1 to the latest Drupal 7.x version, you do not have to apply all the updates that have been released between the versions. You can jump directly from 7.1 to that version. See Update Drupal Core.
Major upgrade procedure
A major release upgrade requires you to first update to the current minor release prior to applying the major release update. If 6.19 is the current version of Drupal 6, and your site is running 6.13, you would first need to update to 6.19 and then apply the update to the current major version 7.
You cannot skip major releases when upgrading your site. This means that if you want to upgrade a Drupal 5 installation to Drupal 7, you must first upgrade from Drupal 5 to 6, and then up to Drupal 7.
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]
1. Make a backup of your Drupal instance.
2. Download the latest release of your current Drupal version.
3. Extract the [tar ball or zip] Drupal package.
4. Set your site on maintenance mode (Found on the Configuration menu in the Admin).
5. Delete all the files & folders inside your original Drupal instance except for /sites folder and any custom files you added elsewhere.
6. Copy all the folders and files except /sites from inside the extracted Drupal package [tar ball or zip package] into your original Drupal instance.
7. 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)
8. If you modified files such as .htaccess or robots.txt re-apply those changes to the new files.
9. Login to your site as administrator or as user no 1
10. Run update.php by navigating to http://...yourdrupalsitename/update.php
11. Follow the process to update your Drupal instance
12. Disable maintenance mode
2. Localization update - An example.
A Norwegian site: