You want to use scp or sftp. These are programs that use ssh to copy files. Scp works pretty much just like cp does and sftp works like an ftp client would.
An scp example, if I wanted to copy a file from my machine (homer.jtshaw.com) to my server (mrburns.jtshaw.com) I would do this: "scp file firstname.lastname@example.org
:~/.". This would copy file from my desktop homer to the root of my home directory on smithers.