This article covers three possible scenarios that may apply to you:
The answer to this problem is almost always to make a new site, in WordPress. Just follow this guide, and set up a website. Then paste all the pictures and text from your old site into the new one. From my experience, this is by far the easiest route 90% of the time.
If the old site is a normal HTML website someone built in a web page editor, then you could download it, and upload it to FatCow's servers. However, unless you know how to use a web page editor, you won't be able to edit your site. You should only go this route if the existing site is very high quality, very large, or you're technically adept, and want to learn how to use a web page editor.
WordPress does have an Import feature (in the Tools menu), which specifically enables the importation of sites made in Blogger, LiveJournal, Movable Type and TypePad, Tumblr, and WordPress.
Unless a specific Import feature exists, a website can't be transferred from one type of system to another. For instance, you can't get a site a web designer built, or one that you made in an "easy site builder" somewhere, and convert it into a WordPress site.
If you have the username and password for your old domain name, then FatCow can transfer it to FatCow's system for you.
However, unless your old site is getting a significant number of visitors, or you have a special country-specific (like .ca or .au) domain name you need to keep, or you just really want to keep your old domain name, you should probably just sign up for a new domain name when you sign up with FatCow. It's much easier.
If you've transferred the domain name to the new site, the old site will no longer be accessible.
If you've abandoned the old site, it will still be there. However, the web hosting or domain name for that site will expire within a year when the bill goes unpaid, and the site will disappear. Don't worry about the site being there. Without promotion and work, it's unlikely that many people will find it.
If you still have control of the old site, it's ideal to leave a "we have moved here" link on it, that points to the new site.