Download the desired version of WordPress from www.wordpress.org to your local computer and unzip the file.
Connect to your website with your favorite FTP client (FileZilla is popular and easy, but any FTP client will be fine).
Upload the folders (and their contents) titled "wp-admin" and "wp-includes" to their matching directories on your server. Be sure to overwrite the current folders.
Upload the individual files in the home folder (index.php, wp-*.php, etc).
Visit and log into your website in order to perform any required database updates.
Note: This method will create orphaned files that can/will build up over time and may present security risks. Be sure to do a file comparison after completion and delete old files off your server from previous WP versions which are no longer in use.