In a nutshell, SSH is a protocol that allows a client computer to connect securely with a server to log in or copy files. It provides authentication and a secure transport layer that allow a user to establish a secure connection even over an otherwise unsecure network.
There are a number of implementations of SSH, but the most popular are the original SSH and OpenSSH. SSH is produced by SSH Communications Security, which was founded by the original developer of SSH, Tatu Ylonen. On the other hand, OpenSSH is part of the OpenBSD project and was developed from the last open source version of SSH.
On Unix systems, both of these implementations usually include a client program called "ssh" and a server program, or daemon, called "sshd." Some client-only implementations exist for Windows and older versions of the Mac OS, while Mac OS X has native versions of the OpenSSH client and server programs.
In addition to the basic client and server programs, most SSH distributions also include a few other tools that allow secure file transfer, such as secure copy (scp) and secure FTP (sftp).