For transferring files between two remote systems (aka through an ssh connection), you can use sftp (Secure File Transfer Protocol).
First, navigate to the folder that you would like to send or receive information from. Once sftp is initiated, the folder for the origin computer cannot be changed. Then, start the protocol with

$ sftp username@place

You can specify the folder to navigate to, if you want, but do not use the ~ in place of /home/username (it will return an error).

$ sftp person@computer.location.org:/home/person/path/to/folder/
Password:
Connected to computer.location.org.
Changing to: /home/person/path/to/folder/
sftp>

You can create new directories in the receiving end normally with mkdir, and use cd as normal. To send information, use put and to receive use get.

sftp> put filename
sftp> get filename

Filezilla is a helpful program for doing sftp through a graphical interface. When using Filezilla, you enter your username and password for where you’re trying to connect. Typically, you will use port 22 for connections.

Space for username, password, and port in the Filezilla QuickConnect 
window.
Setting up Filezilla.