A standard telegraph key has a built-in knife switch that can be used to short the keys contacts. When the key was not in use, that switch was kept closed, so that a signal was continually sent. If that steady-state signal was interrupted, it indicated one of two things: either the operator was about to start transmitting, or something else had happened to break the connection -- such as hostiles cutting the telegraph line.
As the teletype came into use, a key was created that would temporarily interrupt the communications line, allowing this practice to continue. When this occurred, the receiving teletype mechanism would activate but not print characters (it was typing the non-printing "DEL" character repeatedly). The noise got the operator's attention.
Time-sharing computers connected to modems could recognize the break because the "DEL" characters received had "bad parity". This was usually used to generate an interrupt and allow login or stop a running program.
On personal computers, the break key is used in different ways by different programs, but usually involves some sense of changing the interaction the computer, such as switching between multiple login sessions, terminating a program, or interrupting a modem connection.
In programming, especially in old DOS-style BASIC and Pascal, Break is used to stop program execution.
See also SysRq, scroll lock.