This section was adapted from Text-Terminal-HOWTO.
A terminal server is something like an intelligent switch that can connect many modems (or terminals) to one or more computers. It's not a mechanical switch so it may change the speeds and protocols of the streams of data that go thru it. A number of companies make terminal servers: Xyplex, Cisco, 3Com, Computone, Livingston, etc. There are many different types and capabilities. Another HOWTO is needed to compare and describe them (including the possibility of creating your own terminal server with a Linux PC). Most are used for modem connections rather than directly connected terminals.
One use for them is to connect many modems (or terminals) to a high speed network which connects to host computers. Of course the terminal server must have the computing power and software to run network protocols so it is in some ways like a computer. The terminal server may interact with the user and ask what computer to connect to, etc. or it may connect without asking. One may sometimes send jobs to a printer thru a terminal server.
A PC today has enough computing power to act like a terminal server except that each serial port should have its own hardware interrupt. PC's only have a few spare interrupts for this purpose and since they are hard-wired you can't create more by software. A solution is to use an advanced multiport serial card which has its own system of interrupts (or on lower cost models, shares one of the PC's interrupts between a number of ports). See Serial-HOWTO for more info. If such a PC runs Linux with getty running on many serial ports it might be thought of as a terminal server. It is in effect a terminal server if it's linked to other PC's over a network and if its job is mainly to pass thru data and handle the serial port interrupts every 14 (or so) bytes. Software called "radius" is sometimes used.
Today real terminal servers serve more than just terminals. They also serve PC's which emulate terminals, and are sometimes connected to a bank of modems connected to phone lines. Some even include built-in modems. If a terminal (or PC emulating one) is connected directly to a modem, the modem at the other end of the line could be connected to a terminal server. In some cases the terminal server by default expects the callers to use PPP packets, something that real text terminals don't generate.