Wikipedia describes a permalink as
A permalink, or permanent link, is a URL that points to a specific blog or forum entry after it has passed from the front page to the archives. Because a permalink remains unchanged indefinitely, it is less susceptible to link rot. Most modern weblogging and content-syndication software systems support such links. Other types of websites use the term permanent links, but the term permalink is most common within the blogosphere. Permalink is a portmanteau word made from permanent link. Permalinks are often simply stated so as to be human-readable.
Basically, it's the URL for a web page. The Permalinks Admin Page in gpEasy allows you to remove index.php from the URL/Permalink. As noted on the permalinks page, mod rewrite must be enabled on your server.
What's mod rewrite? Let's just say its a way to tell a server how to rewrite a URL. The mod rewrite code is placed into a file named .htaccess.
Mod rewrite is a very advanced and complex topic. I'll post an advanced tutorial here showing the mod rewrites I use with gpEasy to remove index.php as well as www; however, for the beginner you can try to set permalinks in gpEasy but if it doesn't work set it back to the default and leave it alone till you know more.
gpEasy only has two options for permalinks at this point, show or hide index.php. Most people like to hide index.php because the URL looks more readable and it makes working with your internal links a little easier, but that's all it does. It doesn't really help in any other way. It's more of a cosmetics thing than anything.
You can find the permalinks page in the "Admin" menu at the top of the screen or from the main admin page. That's it for this tip.