An idiot light is a warning light (normally, but not necessarily, in an automobile) that replaces a much more useful gauge. Technically the term does not apply to warning lights that do not replace gauges, so a 'doors open' light is NOT an 'idiot light'.
The term arose at about the same time that car manufacturers started replacing gauges with lights - in the late 1960s. Younger people who haven't driven old cars probably don't remember the days when cars came with full instrumentation - my car has a temperature gauge, a fuel gauge, a speedometer, an oil pressure gauge and an ammeter, and some cars had more. By the 1970s, the ammeter was replaced by a battery warning light, the oil pressure gauge by an oil warning light, and the temperature gauge by a temperature warning light on the vast majority of cars. All a modern car comes with is a speedometer, a fuel gauge and, on performance models, a tachometer (rev counter). Everything else is a light.
To those used to the older style of full instrumentation, replacing them with mere warning lights was insulting (thus the term) and also gave less information to the driver. If you know how to read them, gauges can indicate a problem way before things get bad enough to trigger a warning light.
The problem is, of course, that most people don't know how to read gauges and aren't ever going to learn -- quite a few drivers clearly have trouble with the basics of driving, let alone having time to look down and monitor a bunch of different readings and remember them so that they can watch for variation. Marking a safe zone on the gauge helped, but even then most people don't notice.
Gauges are also much more complicated items than warning lights, much more prone to failure, and use up a lot of room - space desired by car interior designers for both larger and easier to read primary instruments such as the speedometer, and for more switches, knobs, dials and cupholders. They also cost more, and that's a problem for the very cost conscious automotive industry.
Nowadays, the only cars that come with anything close to full instrumentation are performance cars - and even then, that's because buyers in this market segment like complexity.
Idiot lights may not do their job. I had a car whose engine was totally ruined by the absence of oil - the drain plug wasn't properly replaced after an oil change - but the 'low oil' light never came on, even after the engine was severely damaged.
The check engine light on modern cars is technically not an 'idiot light' in the true sense, since it does not replace a gauge - it instead is a generic warning that the engine control unit has detected a fault somewhere. It's combined with a second reading, the fault code, that describes what went wrong - unfortunately, auto manufacturers, wishing for a 'no user serviceable parts inside' car, don't let the owner find this out.