An LED cannot have more than its forward bias voltage across it. For an old red LED that voltage is about 1.6 volts. Sacrifice one by driving one with a variable power supply. It will start lighting at a little over 1.5 v. It will be brightest at 1.6v. Only when it burns out will the voltage go much higher than that.

In a circuit, the current is limited by the impedance (resistance) of the circuit driving it. If you drive the LED directly from a i/o pin, the current will be limited by the impedance of the output circuit. The output transistor or FET will act as the resistor and limit the current to some value.

There are two different limits on the current through the LED, the maximum instantaneous current and the max average current. If you exceed the first, the LED will fail immediately. If you exceed the second, it will overheat and die more slowly. There should be a resistor to limit the current to below the max instantaneous current. You can then use PWM to stay below the max average current.

The instantaneous current through the LED can be calculated using i = e / r where r equals the sum of the impedance of the driver plus any external resistor. Heat will be lost by the voltage drop across the driver transistor and the external resistor.