Voltage divider
Two resistors in series between a voltage source and ground, with the output voltage taken at the midpoint. V_out = V_in · R₂/(R₁+R₂). The simplest non-trivial circuit.
Definition
A voltage divider is two resistors in series — R₁ connected to the supply voltage V_in, R₂ connected to ground, and the output taken at the node between them. Kirchhoff's voltage law gives V_out = V_in · R₂/(R₁+R₂): the output voltage equals the input scaled by the ratio of the lower resistor to the total. Set R₁ = R₂ for half the input; set R₁ much greater than R₂ for a small fraction; set R₁ ≪ R₂ for nearly the full input.
The catch is that the formula assumes no current is being drawn out of the midpoint. Connect a load R_L to V_out and R₂ is effectively replaced by the parallel combination R₂ ∥ R_L, which shifts V_out. This is why voltage dividers are used as reference generators only when the load has high input impedance (voltmeters, op-amp inputs, MCU analogue-to-digital pins). For power delivery, a regulator or buffer is needed.
Dividers are everywhere. They set bias voltages for amplifier transistors, scale sensor outputs to fit an ADC range, provide reference voltages for comparators, divide high-voltage rails down for measurement, and build the resistance ladder of an R-2R digital-to-analogue converter. The potentiometer — a variable resistor with a sliding contact — is a voltage divider with a continuously adjustable ratio. Every volume knob, every joystick, every slide potentiometer is this circuit.