In the event that you want to detect a voltage passing a threshold, rather than being merely HIGH or LOW, you can use the analog comparator. Example sketch:

volatile boolean triggered;

ISR (ANALOG_COMP_vect)
  {
  triggered = true;
  }

void setup ()
  {
  Serial.begin (115200);
  Serial.println ("Started.");
  ADCSRB = 0;           // (Disable) ACME: Analog Comparator Multiplexer Enable
  ACSR =  bit (ACI)     // (Clear) Analog Comparator Interrupt Flag
        | bit (ACIE)    // Analog Comparator Interrupt Enable
        | bit (ACIS1);  // ACIS1, ACIS0: Analog Comparator Interrupt Mode Select (trigger on falling edge)
   }  // end of setup

void loop ()
  {
  if (triggered)
    {
    Serial.println ("Triggered!"); 
    triggered = false;
    }
  
  }  // end of loop

This can be useful for things like light-detectors, where you might need to detect a change from (say) 1V to 2V on an input.

Example circuit:

You can also use the Input Capture Unit on the processor, which will remember the exact time of certain inputs, by saving the current count of Timer/Counter 1. This lets you store the exact (well, almost exact) moment that the event of interest occurred, rather than introducing the delay (of probably a few microseconds) before an ISR can be used to capture the current time.

For timing-critical applications, this can give somewhat increased accuracy.

Example application: Turn your Arduino into a capacitor tester