The file /dev/ptmx is a character file with major number 5 and minor number 2, usually of mode 0666 and owner.group of root.root. It is used to create a pseudo-terminal master and slave pair.
When a process opens /dev/ptmx, it gets a file descriptor for a pseudo-terminal master (PTM), and a pseudo-terminal slave (PTS) device is created in the /dev/pts directory. Each file descriptor obtained by opening /dev/ptmx is an independent PTM with its own associated PTS, whose path can be found by passing the descriptor to ptsname(3).
Before opening the pseudo-terminal slave, you must pass the masters file descriptor to grantpt(3) and unlockpt(3).
Once both the pseudo-terminal master and slave are open, the slave provides processes with an interface that is identical to that of a real terminal.
Data written to the slave is presented on the master descriptor as input. Data written to the master is presented to the slave as input.
In practice, pseudo-terminals are used for implementing terminal emulators such as xterm(1), in which data read from the pseudo-terminal master is interpreted by the application in the same way a real terminal would interpret the data, and for implementing remote-login programs such as sshd(8), in which data read from the pseudo-terminal master is sent across the network to a client program that is connected to a terminal or terminal emulator.
Pseudo-terminals can also be used to send input to programs that normally refuse to read input from pipes (such as su(8), and passwd(8)).
The Linux support for the above (known as Unix98 pty naming) is done using the devpts filesystem, that should be mounted on /dev/pts.
Before this Unix98 scheme, master ptys were called /dev/ptyp0", ..." and slave ptys /dev/ttyp0", ..." and one needed lots of preallocated device nodes.
getpt(3), grantpt(3), ptsname(3), unlockpt(3)