It's called scp. Where SSH stands for "secure shell", SCP stands for "secure copy", they use the same port and the same protocol. I don't believe putty can do it, but there are other Windows programs that can. WinSCP is one, or Cygwin will provide you with a Linux-like CLI that you can scp to/from as well.
Usage is simple:
scp localfile user@host:/path/to/remotefile
to send localfile to the remote host, or
scp user@host:/path/to/remotefile localfile
to retrieve remotefile from the host and save it to localfile