The title & the gratis tag say it all.
Have a look at Baby FTP Server.
This Baby FTP server has only the most necessary features and is yet powerful enough to be a basis for a more complex server.
It's a single executable that requires no admin rights nor installation. Just download, double click and you're good to go.
So, the one I've found is called Quick`n Easy FTP Server Lite. It is pretty simple to use but does give if need quite a few advanced options. It's free and compatible with windows XP. Now, in regard to your main requirement, it can run without admin rights but here's what you need to know: You must allow the port to go through your router (for public use) and window's firewall itself (which usually requires admin rights). There is no way around the windows firewall with getting an open port without admin rights but you may not need them depending on how to firewall was setup (try and see if it works). In any case, this software should be exactly what you're looking for - it even has a XP control panel looking interface:).
Quick `n Easy FTP Server Lite is a standalone, multi-threaded FTP server that offers advanced features and security. The interface is simple and intuitive, with organized sections for managing users, configuration...