Would it be possible to power a small LED light using the headset jack of a regular smartphone, and if so, how would you connect the LED light with an audio jack. How would such an implement look like. I'm not very versed in electrical terminology so simplified explanations are appreciated.
This is a very interesting question. I would say, yes and no. The headphone jack is not designed for this at all, and you won't get much power out of it.
My preliminary research indicates that an iPhone 4 headphone jack may support up to 60 mW of power output. This, however would be AC power, like sound.
So, I think you could put a full bridge rectifier in to make the signal only in the positive direction, then smooth the peaks of the sine wave to a dc signal (approximately) with a capacitor. The you have a DC source. Then it's as simple as running an LED off of a battery. The thing is, you're not going to have much power. Otherwise, I would still have to say it is technically possible.
By the way, to turn it on you would basically just be playing a sine wave beep. To turn it off, just stop playing the sound.
Can I ask what you were hoping to do with this LED? I don't think you could make a very good flashlight with it, if that's what you mean.