Go to your Cpanel, enter to file manager, navigate to public_html folder, select all the files and folders, and use the compress button.
Download the zip file.
After that, go to phpmyadmin in the cpanel, click in your database nave in the right column, go to export in the top menu, and export the file compressed or not, anything will be fine.
Now you have your wp files and the db, you can later up this anywhere else with a couple of modification.
To have a complete copy of everything, you’ll need to back up both the “files” and the “database” for your WordPress site.
There are a number of ways to accomplish this.
One rather straightforward method is to use a plugin like WP-DB-Backup to turn all of your database content into “files” on your web host.
Then, you can connect to your web host via FTP and copy all of the files to your computer. If you’ve first backed up the database to your files (the plugin I mentioned will place the database backup inside a folder in the ‘wp-content’ folder of your WordPress files), then by downloading all of the files you will effectively be backing up everything: your entire WordPress installation (settings, theme, etc), content (posts, pages, etc) and uploads (images, files, etc.).
One caveat here: the process above makes a true backup of WordPress. It won’t be particularly useful without “restoring” it somewhere (e.g. to another web host). It is possible to set up a local installation of WordPress on your computer, however, if you need to have access to all of this. That’s a bit more advanced, however.
Just as I was about to publish this answer, I re-read your question. It made me realize that there are a few possibilities about how your blog is configured. So let me clarify: my answer (above) assumes that your WordPress blog is actually hosted in your HostGator account. You mention that HostGator is your “website” host. For many of us, that means that your website is built on WordPress. If that’s the case, then this answer will apply.
If, however, your “website” and your WordPress blog are two different things, then we’ll need to know more about how everything is set up.
Looks like Forensica Digital and I were both typing up answers at the same time. Either method will give you a copy of both your database and your files.
