Visit Joomla.org

Visit joomla.org and click on the black "Download Joomla" button on the right.
Download Joomla

Download the Joomla files that you need by clicking on the "ZIP" link. In this case, we'll be using the latest version.
Login to Your Cpanel

Often you can access your CPanel just by adding /cpanel to your domain name ... example.com/cpanel
Access Your File Manager

Although your screen may not look 100% identical to the one above, every version of Cpanel does come with a "File Manger" option somewhere.
Go to Your Web Root

Visit the Upload Area

If You Want to Overwrite Existing Files

If you're looking to overwrite existing files, be sure to make a backup of your configuration.php file. You'll need that for your site to continue to run properly.
Upload Your Files

Hit "Browse" and select the Joomla files that you downloaded earlier.
Extract Those Files

Extract the Joomla files using the "Extract" button.
Final Steps

If you want to replace your existing Joomla site and keep your site running, you'll now need to do two more things:
- Re-upload your configuration.php using the configuration.php process as before
- Delete the installation folder as per the image above