microphone to a microcontroller. very simple
What microphone are you using? What microcontroller? What is your sound source? We are going to need that information to help you build an actual circuit that you can use for this application. As some of the comments said, this doesn't have to be done with a microcontroller, but it will probably make things easier to understand starting off.

As a general idea, your microphone on its own might not have enough of a voltage swing for you to read it directly into the ADC of a uC. You'll want to use an op-amp to amplify the signal coming out of the microphone (you might need to bias the mic as well, depending on specifics of what you're working with). Once you read in the output of the amp in your ADC, your program can decide whether or not to turn on the LED.

If you want to do it without a microcontroller, you could send the output of the microphone amplifier into a comparator and use a potentiometer to adjust when the comparator triggers high, driving an LED. This gives you a physical way of adjusting how loud the sound needs to be before the LED turns on, which is a pretty neat experiment to start off doing as well.