How does a zener diode and a resistor regulate voltage?

Let's assume your input voltage is 10V, the zener is a 5V type and the resistor is 100\$\Omega\$. The zener will set \$V_{OUT}\$ to 5V, so that the current through the resistor will be \$I = \frac{10V - 5V}{100\Omega } = 50mA\$. When you put a load on the output, say 500\$\Omega\$ the zener will still keep the output at 5V, so R will still have 50mA through it, but part of that current will go through the load: \$I_L = \frac{5V}{500\Omega} = 10mA\$, so that there's only 40mA through the zener. For lower load resistances the current though the load will increase, taking away current from the zener diode, until it has too little current left to regulate \$V_{OUT}\$ properly. The 50mA from the example wasn't chosen at random; it's often the current at which the zener voltage is specified (especially for older zeners). More modern zeners can work at currents far below 1mA. If \$R_L\$ is less than 100\$\Omega\$ the resistor divider \$R/R_L\$ will pull \$V_{OUT}\$ below the zener voltage, the zener won't play anymore and \$V_{OUT}\$ will drop below 5V.
So far for load regulation.

Line regulation tells how a regulator reacts to variations in input voltage. Let's take our example with the 500\$\Omega\$ load, and decrease the input voltage to 9V. The output voltage is still kept at 5V by the zener, so the current through the load remains 10mA, but the current though R will be \$I = \frac{9V - 5V}{100\Omega } = 40mA\$, and therefore the current through the zener 30mA. Again the input voltage can decrease to the point where the zener has too little current left to operate properly. The upper limit of the input voltage is determined by the maximum current allowed through zener and R.

This type of voltage regulation is very simple, but not very good. Line regulation is poor, which means that the output voltage will still vary a bit when the input voltage increases/decreases. Same with load regulation: output voltage will vary with varying loads. And compared to the maximum load there's a rather big loss in the zener, so it's not very efficient. A small integrated regulator like an LM78Lxx is always a better choice.

Look at the Zener diode curve. You will see that the device breaks down at the Zener voltage when reverse-biased, and conducts. That property will fix the output voltage at the breakdown voltage, over a range of output currents, when used with a resistor, with relatively small voltage changes. It will also stabilise the output against changes in the input voltage.

Strictly speaking, Zener diodes are low-voltage devices (up to about 5V6). Higher-voltage ones have a different mode of operation and are called avalanche diodes. Both types are commonly referred to as Zeners, though.