Electrical switch/relay that will stay on once triggered (regardless of future inputs)
What you describe is simple latching behavior.  There are lots of ways to get that.

You can wire up a relay so that it keeps itself energized once it is energized.  A SCR is a electonic device that inherently latches once triggered, and stays on as long as some minimum current is supplied.  There are various ways of wiring up transistors to do this.  Basically anything with a gain greater than 1 can be arranged to latch.  There are digital gates specifically for this purpose, and of course a microcontroller can be trivially programmed for this function.