How do I use the timers on an Arduino Uno R3?
Your question is probably not as simple as it may seem.

One approach I would use if I was to address this same problem would consist in:


converting the analog output of the mike into a digital value (based on a threshold to determine)
connect this digital signal to one Arduino pin that can trigger an interrupt when its level changes


Let's first start with the easy part: Arduino program.

Let's say we have a digital signal that raises to 1 as soon as a sound of too high amplitude is captured by the mike; it shall stay 1 as long as the sound does not reduce amplitude enough. We will connect this signal to Arduino pin 3; then we can use attachInterrupt() to have a function called whenever this pon level raises to 1:

volatile static bool started = false;
volatile static unsigned long timing = 0;
volatile static bool finished = false;

void loud() {
    if (started) {
        timing = micros() - timing;
        started = false;
        finished = true;
    } else {
        timing = micros();
        started = true;
        finished = false;
    }
}

void setup() {
    pinMode(3, INPUT);
    attachInterrupt(1, loud, RAISE);
}

void loop() {
    if (finished) {
        // Do something with timing
    }
}


Now the hardest part: electronics!

A first issue we have to deal with is related to the fact that the raw mike signal will have same wave shape as the sound it captures, this wave has a frequency that can vary between 10Hz (bass) and 20KHz (treble).

Indeed, if we convert the mike analog output to a logical value (1 when the analog value is above a threshold we'll call Vt, 0 when it is under Vt) fast enough, then, if the environmental sound as an amplitude above Vt, the logical input connected to your Arduino will oscillate at the same frequency as the sound! This would make our efforts pointless as you would measure the period of the sound wave instead of the elapsed time between two distinct sound bursts.

That means that we should probably first "smoothen" the analog sound signal in order to avoid disturbance of its frequency.

Hence point 1 will have to deal with these 2 aspects:


smoothen analog input value
compare smoothened value with Vt threshold and provide a logical value from this comparison


Comparison can rather easily be performed by an op-amp used as a comparator.

However, the main difficulty you may have with an op-amp is the need for a symetric voltage supply (e.g. +5V/-5V) although it is possible to use a single supply but that raises further limitations (outside the scope of my answer).

Smoothening the signal is potentially more complex, although it can be done with a simple capacitor, because smoothening the signal generally means weakening it, thus you would lose the initial amplitude of the sound signal.

So how can we deal with that? 

A Schmitt trigger is probably a good solution for this problem but further experiment will be needed to confirm that.