ssh - Secure SHell - like telnet (and rlogin and rsh), is a way to log on to a remote host and execute commands on it. Unlike telnet - where usernames and passwords are sent in clear text and are exposed to anyone running a packet sniffer on the network, ssh, encrypts all communication between you and the remote host - thus ensuring privacy. ssh also uses public keys for identification - to ensure that you are who you say you are, and the remote host is who it says it is - thus ensuring authenticity.
To use ssh, the remote host must be running sshd (the ssh daemon) and you would use ssh instead of telnet. (you also need to set up your public- and private- keys before hand - but I won't go into that)