What is a decoupling capacitor (or smoothing capacitor as referred to in the link below)?

How do I know if I need one and if so, what size and where it needs to go? 

This question mentions many chips needing one between VCC and GND; how do I know if a specific chip is one? 

Would an SN74195N 4-bit parallel access shift register used with an Arduino need one? (To use my current project as an example) Why or why not?

I feel like I'm starting to understand the basics of resistors and some places they're used, what values should be used in said places, etc, and I'd like to understand capacitors at the basic level as well.

I was the one that asked that question. Here is my rudimentary understanding:
You attach capacitors across \$V_{CC}\$/GND to try to keep the voltage more constant. Under a DC circuit, a capacitor acts as an open circuit so there is no problem with shorting there. As your device is powered up (\$V_{CC}\$=5V), the capacitor is charged to capacity and waits until there is a change in the voltage between \$V_{CC}\$ and GND (\$V_{CC}\$=4.5V). At this point, the capacitor will discharge to try to bring the voltage back to the level of charge inside the capacitor (5V). This is called "smoothing" (or at least that is what I call it) because the change in voltage will be less pronounced.
Ultimately, the voltage will not ever return to 5V through a capacitor, rather the capacitor will discharge until the charge inside it is equal to the supply voltage (to an equilibrium). A similar mechanism is responsible for smoothing if \$V_{CC}\$ increases too far beyond its average (\$V_{CC}\$=5.5V perhaps).
As for why you need them, they are very important in high-speed digital and analog circuits. I can't imagine you would need one for an SN74195, but it can't hurt!
