Divider. To understand what happens in this circuit of three elements, first of all, you have to have an idea about such fundamental electrical concepts as voltage divider and current divider.
Dynamic resistor. Then, you have to have an idea about the diode behavior as a voltage-stable nonlinear element no matter what the specific diode is (LED, Zener, Si, Ge, etc.) You can achieve this in a very simple way by thinking of the diode as of a dynamic resistor. You can even mimic it by a humble variable resistor (rheostat) as I have explained in my answers to similar questions below:
Why does the voltage drop across an LED matter when determining an appropriate resistor value?
How is a series resistor limiting the voltage for a diode?
Art of Electronics - Zener Diode Example
Zener diode can vary current flow to maintain voltage drop, how does this magic effect work?
Why does voltage drop in a simple zener diode based voltage regulator?
What is the main difference between a zener diode and normal diode?
Understanding why not to use a resistor for multiple LEDs
What are the disadvantages of using a Zener diode over a linear voltage regulator?
As you can see from these explanations, the diode clever trick is extremely simple - it acts as a variable resistor that changes its resistance R in an opposite direction to the current I flowing through it. As an example, when we increase the current, the diode will decrease its resistance and vice versa, when we decrease the current, the diode will increase its resistance (here I mean "resistance" in the broadest sense of the word - as something that impedes the current). As a result, the product of the two variables - the voltage drop V = I.R across the "resistor", stays constant.
Dynamic voltage divider. If we now connect a resistor (R1) in series to the diode, we will get a dynamic voltage divider. When we increase the supply voltage, the diode will decrease its resistance and vice versa, when we decrease the voltage, the diode will increase its resistance. As a result, the voltage drop across the diode will stay constant.
Dynamic current divider. If we add another resistor (R2) but now in parallel to the LED, we will get a dynamic current divider. Now, if we increase the supply voltage, the diode will decrease its resistance and will divert a part of the current flowing through R2... and vice versa. As a result, the voltage drop across the diode will stay constant as above.
The operation of the circuit of a resistor and diode in series can be graphically illustrated by superimposing the IV curves of the two elements (the so-called "load line" technique).
I have added the IV curve of the variable resistor to show the mechanism of the dynamic resistance (see my explanations in the links above).
This graphical representation can be even used for calculation (and it was widely used in the past since the analysis of such non-linear circuits is complicated).
When another resistor R2 is connected in parallel to the diode (voltage divider), we have to combine two IV curves - R2||LED or R1||R2 (Thevenin).
In the answers and comments here it was convincingly shown that the voltage divider should not be used to supply diodes (LED). However, there are cases when this is desired. Here are some of them.
1. For example, in an LED threshold voltage indicator, we want the LED to light up at some voltage value. In this case, the voltage divider will help to adjust the threshold (the single resistor cannot). Let's consider the circuit operation when the input voltage increases from zero to maximum.
As long as the output voltage of the voltage divider is less than the LED threshold voltage, the divider is unloaded (open circuit)... and the LED is controlled by voltage. When its threshold voltage is reached, the LED turns on and shunts the lower resistor R2. Now the LED is controlled by current set by R1 and the input voltage.
2. Another example is an LED connected to the collector of a transistor. We don't want it to glow when the transistor is off ... but it may glow dimly due to the small residual current of the transistor. Then it helps to connect a resistor in parallel with the LED. This is what I had to do years ago when I demonstrated to my students the idea of a transistor latch.
3. The same problem arises when we want to turn off a BJT by interrupting its base current. Then we have to connect a resistor in parallel to the base-emitter junction. See the explanations in my answer to a related question.
4. In a common-emitter amplifier stage, they prefer to bias the base-emitter junction by a voltage divider (by voltage) than by a single base resistor (by current). Thus the bias voltage is less affected by the temperature; so the operating point is more stable.