How would I copy somefile.zip to another directory?
If you are connected to a shell session via ssh, and the file exists on the remote system, you can use the cp command.
cp sourcefile destfile
If the file exists on the local system you should use sftp. If your local machine runs Windows, WinSCP is an excellent sftp client.
Assuming you're using the command line then like this
scp myfile.txt dest/
for a local file or alternatively
scp user@myserver:/path/to/myfile.txt user@myserver:/path/to/dest/
If you are using Win32, Tunnellier is a reasonable solution which not only allows file transfer, but secure remote access as well.