First things first: ssh is a way to remotely login to another computer. The shell (command line) you get after you ssh is (pretty much) the same as if you had opened a xterm in the remote machine. If offers no such way to move files.
However, the fact that the remote computer accepts ssh connections gives you some options to exchange files:
Use scp
To copy from your local computer to the remote, type, in the local computer:
scp /tmp/file user@example.com:/home/name/dir
(where /tmp/file can be replaced with any local file and /home/name/dir with any remote directory)
To copy from the remote computer to the local one, type, in the local computer:
scp user@example.com:/home/name/dir/file /tmp
Use sshfs
This is a little more advanced but much, much nicer (when the internet connection of both computers is good. If not, stick to scp)
You can "link" a directory from the remote computer to an (empty) directory of the local computer. Say you "link" the /some/remote/dir from the remote computer to /home/youruser/remotecomp in your computer. If there is a file /some/remote/dir/file in the remote computer, you can see it on /home/youruser/remotecomp/file. You can copy and mv as usual, and you can even alter remote files and dirs.
Note however, that when the connection ends, /home/youruser/remotecomp becomes an empty dir again, and you only keep in the local computer the files you copied to other directories
To achieve this:
1) install sshfs:
sudo apt-get install sshfs
2) create a empty dir
mkdir /home/youruser/remotecomp
3) "link" the two directories (the right term is mount)
sshfs user@server.com:/some/remote/dir /home/youruser/remotecomp
4) Enjoy
5) "unlink" the dirs
fusermount -u /home/youruser/remotecomp
If the local computer runs windows
You can find versions of scp
for windows. See, e.g, winscp