Electronic – How to set PSU voltage using Arduino

Many supplies nowadays use both a switching regulator and a linear regulator together. The two are set up to track each other, so the voltage dropped by the linear regulator is never excessive.
So for example, say you set the supply to 5V, you might have a 24V input which the switching regulator drops down to 6-7V, then the linear regulator drops that down to the desired 5V. This way you get the benefit of the switchers efficiency with the quiet output of the linear regulator.

However, as mentioned by Passerby, efficiency and weight is not always so much of a concern compared to reliability and quietness, so many simply use a purely linear supply and deal with the losses involved.
To keep from being ridiculously inefficient however, generally many linear supplies will have more than one tap on the transformer which are switched between at appropriate points in the range (e.g. instead of dropping from 24V to 1V, it would switch to, say, a 5V tap so a lot less power is dissipated) You can usually hear the clicks of the relays at certain points in the voltage range.

Each of the 3 power rails (3.3V, 5V, +12V) are connected to a fuse, then to a binding post, while the negative binding posts go to ground.

This sounds like the problem. Keep in mind, voltages are differences in potential between two points. There's nothing special about ground, it's just an arbitrary point which we pick. It's 0V because the difference of something with itself is 0.

You can call anything you want "ground". That's why this circuit works. If you call the -12V output as "ground", then everything else is 12V higher. That's including what was previously called "ground": now it's 12V, because it's 12V more than what you are now calling ground.

Now consider what you've done:

^{simulate this circuit – Schematic created using CircuitLab}

The "ground" of the power supply connects all of the voltages supplied together (connections labeled B and C). The output voltages are relative to this. Notice how the -12V (V4) makes a negative voltage because it's positive side is attached to "ground".

Then, you attached the negative binding posts of all the supplies together. Largely this is redundant: you are duplicating connections B and C. But you are also adding connection A.

See the problem? You've shorted out V4. A wire has ideally zero resistance. By Ohm's law, the current that will flow is:

$$ \frac{12V}{0\Omega} = $$

In reality, the wires used to make this connection actually have some very small resistance, and you get a whole ton of current. This far exceeded the current the voltage regulator can handle and the smoke got out.