SSH is a protocol which allows to to login to a text command prompt on one of the remote servers. This allows you to start commands directly on that server. If your desktop is not Linux based, you will need a terminal emulation prgram to create that connection.
On the remote server you can use the SFTP command to transfer files from one server to the other. SFTP is a command line tool which allows you send files from one of the servers to another.
I use 'scp' (secure copy) to copy files. Just like copy, but encrypted and between two servers. It's the only thing I use, never use FTP because I don't run ftp on my machines.
You may also want to add the -C switch, which compresses the file. That can speed things up a bit. i.e. copy file1 from server1 to server2, on server1: scp -C /home/user/file1 firstname.lastname@example.org:/home/user
You must have ssh running on server2 to make this work though.
If you want to automate the copying, say through a cron job running on server1 (so it copies files every night) you may want to use rsync instead of scp. rsync is like scp, but adds the capability of only copying over changed files, thus reducing bandwidth. I use rsync to automate copying my webserver backups to an offsite machine. Full details here: [webmasterworld.com...]
The important part in that post is setting up the public keys so that server1 can log into server2 without getting prompted for a userid/password. However the same thing works for scp as well - if you set up the keys as noted in that post, then server1 can scp or ssh (or rsync) to server2 without a password request.