When would I use a voltage regulator vs voltage divider?
These two circuits types have very different applications.
A resistor divider is generally used to scale a voltage so that it can be sensed/detected/analysed more easily.
For example, say you want to monitor a battery voltage. The voltage may go as high as 15V. You are using a microcontroller's analog-to-digital converter ("ADC"), which is using 3.3V for its reference. In this case, you may choose to divide the voltage by 5, which will give you up to 3.0V at the input of the ADC.
There are a couple of drawbacks. One is that there is always current flowing through the resistors. This is important in power-constrained (battery powered) circuits. The second problem is that the divider can't source any significant current. If you start drawing current, it changes the divider ratio, and things don't go as planned :) So, it's really only used to drive high-impedance connections.
A voltage regulator, on the other hand, is designed to provide a fixed voltage regardless of its input. This is what you want to use to provide power to other circuitry.
As far as creating multiple voltage rails: For this example, let's assume that you are using switching regulators that are 80% efficient. Say that you have 9V, and want to produce 5V and 3.3V. If you use the regulators in parallel, hooking each one up to 9V, then both rails will be 80% efficient. If, however, you create 5V and then use that to create 3.3V, then your 3.3V efficiency is (0.8 * 0.8) = only 64% efficient. Topology matters!
Linear regulators, on the other hand, are assessed differently. They simply lower the output voltage, for any given current. The power difference is wasted as heat. If you have 10V in, and 5V out, then they are 50% efficient.
They have their benefits, though! They are smaller, less expensive, and less complicated. They're electrically quiet, and create a smooth output voltage. And, if there isn't much difference between the input and output voltages, then the efficiency can top a switching supply.
There are ICs which provide multiple regulators. Linear Tech, Maxim Integrated, Texas Instruments, all have a good selection. The LTC3553, for example, provides a combination of a Lithium battery charger, a switching buck regulator, and a linear regulator. They have flavors with or without the charger, some with two switchers and no linears, some with multiple linears...
One of my current products uses a 3.7V battery, and needs 3.3V and 2.5V. It was most efficient for me to a linear for the 3.3V, and a switcher for the 2.5V (fed by the battery, not the 3.3V rail). I used the LTC3553.
You'll want to spend some time on their respective website's product selector tools.
Good luck!
Since a voltage divider does not regulate, one would not want to use a voltage divider when one wants a regulated voltage.
A voltage regulator will, within its limits, maintain the output voltage at a fixed value even as the input voltage and load current varies.
A voltage divider will not do this. Consider the voltage divider equation:
$$v_{OUT} = v_{IN}\frac{R_2}{R_1 + R_2} - i_{OUT}\cdot R_1||R_2$$
which is manifestly dependent on \$v_{IN}\$ and \$i_{OUT}\$ so a voltage divider is not a voltage regulator.
There are, however, plenty of applications for voltage dividers, e.g., attenuation, but voltage regulation is not one of them.
A voltage divider is particularly bad at providing a fixed voltage to a variable or low-impedance load. Variable loads are quite common, and include most digital circuits on the planet.
Fixed, high-impedance loads can have a voltage divider in front of them. This is the case when using an ADC to measure or a comparator to fence a much larger voltage, or in the sense input of a voltage regulator.