I used to be connect to my Cable modem throw a Linksys router, but now I am connected directly into the cable modem. When I was on the router my hostname was localhost.localdomain. NOw that I am connected directly to the modem its changed... to <number>.<number>.<number>.<number>.cfl.res.rr.com .........
I'm using Fedora Core 3. When I go to System Settings -> Network, the DNS Tab has my hostname still as localhost.localdomain.

Why did this happen ? Can I change it with any ill effects to my cable connection ? If so, how can I change it ?

With the router you were effectively on an internal private network. Now you are hooked up directly to the net ,i.e your ip is the one you cable service provides.

It's normal and no you can't change it, but your box is still localhost. Although now without a router you obviously can't be on your own network so it's pretty much meaningless.

$ hostname

Should still return localhost. Why did you get rid of the router anyway? It provides a great first line of defense.

I didn't get rid of it actually. Me and my roommate share our connection through the router, but I live in Florida and some lightening
had a field day wit some of our electronics since we weren't home to unplug them.

So what if I wanted to change my hostname to something like myhost.mydomain.org ?

You can change it to whatever you want in the /etc/sysconfig/network file.