A fax machine is a device that is used to transmit or receive a copy of a printed document via a standard telephone line. The technology is rather antiquated now. This is most obvious when one notices the speed of a fax transmission - a maximum 9600bps (bits per second). Even a dodgy modem that has been kicked around the floor and covered in jam is usually capable of several times that speed.
At the beginning of the century, faxes were used only by the military, then by bureaus and businesses. Now they are much more commonplace. They are being sold as combination fax/copying machines; and as telephone/fax devices that specialise in deafening the person at the other end of the phone line with a shrill shrieking sound, if they accidentally pick up the receiver when a fax is being sent to them.
Note: if you want to get a fax machine... don't. Get an e-mail account instead. It is much less trouble, less likely to deafen you (unless you have a very loud modem), and you can send and receive colour documents. If you really must get a fax, then make sure you get a plain paper one. The heat sensitive paper used by traditional faxes has a tendency to fade surprisingly quickly, thus losing all your precious documents forever.