There are two ways of doing this.
The Short Method:
1. Log into the backend of your site, and install this extension:
http://extensions.joomla.org/extensions ... I7fQ%3D%3D
2. Under "Components" click on the J!Update button. Follow the instructions. You can make a backup of your site with this component.
3. If all goes well, you should have your site up to speed with the latest version of Joomla 1.5.
The Long Method:
1. Use an FTP program to download your whole website to your computer. If you are unsure of what FTP program to use, download Filezilla, which works really well and is free.
2. On your website space (hosting space), create a new folder, and call it something like "backup".
3. Take the website you just downloaded, and upload it to this new folder. Once all of the files have uploaded (which may take some time), you'll have a duplicate of your site that you can visit at
www.yourdomain.com/backup
4. Upload the latest version of Joomla (except for the "Installation" folder) to your Backup folder. Replace any files it asks you to replace.
5. Once all of the files from Joomla 1.5.26 have been uploaded, test your site in the backup folder. Make sure it all works well.
6. If all is well, you can delete your backup folder, and upload the Joomla package to your main root folder (except for the "Installation" folder, don't ever upload this folder when simply doing an update). That should be it.
Also, public_html is a generic name for the folder that contains your website. Some hosts call this "www" and others call it "root". It just depends on your website host. You can only see this folder if you use an FTP program to connect to your website hosting.