Fedora Core, now just Fedora, is a free and completely open platform based on and supported by Red Hat. The "Red Hat" name is reserved for the Enterprise product.
Fedora intends to be on the bleeding edge of software, making it a great enthusiast platform and also great for people who "wants them features nows". The downside is that there isn't any real long-term support provided by Fedora.

This is the very first version, released in 2003 when the original Red Hat Linux was discontinued.

This is the default GNOME desktop. Fedora Core still uses the old red hat logo, but everything else has been changed to have Fedora branding on it.
The desktop is visually similar to previous Red Hat releases.

Upon trying to change the resolution, it asks me to restart the x server. This is disappointing, considering Windows 95 could happily switch video modes without a reboot, and this is a 2003 Linux we are talking about.

The menu is arranged almost identically as previous versions of Red Hat.

For a file manager Fedora Core uses Nautilus.

Fedora Core uses the Mozilla web browser, although you could presumably install anything you wanted.