TELNET is essentially a protocol for making remote console connections over a TCP/IP network.
By default, a telnet client asks for a destination address (DNS name or IP address), and a port number. If no port is specified, the well known port for telnet is 23, so this is the default.
What happens once a connection is established, depends entirely on what application is listening on the port at the other end! By default, the server that listens on port 23 is the telnet server. This (after exchanging some pleasantries with the client to decide what display protocols to use) displays a login prompt, and then a remote shell for the user. If you have a shell account, this is the traditional way to access it, and then run processes, read E-Mail etc (although see note about security below).
However, there's nothing to stop you running a different server on port 23, or for that matter telnet'ing to a different port - such as 25 (SMTP), 110 (POP3) or 80 (HTTP). Most Internet protocols are text-based, and some will even respond to the "help" command. These things can even be useful...
- Connecting to a remote server's POP3 port allows you to read any incoming mail for you.
- Connecting to a remote server's SMTP port allows you to send E-Mail onwards with some anonymity (although most SMTP servers will at least record your real IP address in the header).
- Connecting to a remote server's HTTP port allows you to see in more detail the HTTP status messages etc which a browser may hide from you.
- and so on...
The other big advantage of telnet is that it's everywhere. Every flavour of Unix and every version of Windows since Windows 95 (although until Windows 2000 they weren't very good versions!) has it.
However, telnet has a major problem. It's not secure - everything is sent in plain text be it over a local network or over the Internet. SSH is a much better solution for this, as everything is strongly encrypted. SSH can also handle encrypted file transfer using SCP and port forwarding. SSH used to cost something, but since OpenSSH has been around, it is available free for Unix platforms, and SSH clients are free on Windows (eg PuTTY).
Update 2012
Oh, the irony. "The other big advantage of telnet is that it's everywhere". Recent versions of Windows don't have telnet installed by default (although it can be added as as feature). I can perhaps understand this for the workstation configurations, but for the Server versions??