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When one polarity is applied, current flows through the LED and it lights up. In this state it will drop voltage across the resistor to maintain a steady voltage across the LED. For indicator LEDs, this is typically around 1.5V and for high brightness types, about 3.5V.
When the reverse polarity is applied, it doesn't conduct or light up. Because it doesn't conduct, no voltage is dropped across the resistor and if it was possible, full voltage would appear across the LED. However, in reverse direction most LEDs have a very low breakdown voltage, it's typically only about 5V. If you exceed the breakdown voltage the LED will conduct but is also liable to being damaged. When a reversed normal diode is connected across the LED in such a way that it conducts when the LED voltage is reversed, it draws current and drops the voltage across itself to around 0.7V, well below the danger level that could damage the LED.
So on one half AC cycle the LED lights and on the other half cycle the diode conducts and protects the LED.
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