That's a picture of the current-voltage graph of a diode - how much current passes through the diode for a specific amount of voltage (or the opposite - the voltage drop per amount of current passing through it!) Any diode (even an LED) has a graph that looks like that, but with different scales. For a resistor, the graph would look like a straight line. Ohm's law just states that this line is linear, and the slope varies with the resistance.
Diodes, on the other hand, are not linear. Anywhere from Vbr to Vd (on the picture) means pretty much no current is passing through (note: Vbr is usually a huge negative voltage, like 500V. That means the diode doesn't act one-way anymore at about 500V. Vd is around 3.3V for an LED and around 0.7V for a silicon diode). When your voltage gets above Vd, the LED "takes up" less than the "available" voltage and then the resistor uses up the rest.
The nearly vertical line after Vd is why you need the resistor when using an LED. If you fed it a voltage without any resistor, either very little current would pass through and not light it up (brightness depends on current times voltage), or a huge amount of current would pass through and heat it up enough to burn it out. It is possible to get the voltage exactly correct to have it light up perfectly, but manufacturing defects on the LED and slight voltage differences on the battery would make that far too risky.