Faucet
a device with a hand-operated valve for regulating the flow of a liquid
Spigot
a faucet
the valve or plug in a faucet
Tap
a device for starting or stopping the flow of liquid in a pipe, barrel, etc.; faucet
They all can be used when you're talking about a device that starts or stops the flow of a liquid, but there are some regional variations in how they're commonly used.
In my region, faucet is used for the common household fixture that can mix hot and cold water together and control how fast the water flows.
A spigot is a single knob faucet that only has one pipe it controls, like the outdoor spigot that you connect a garden hose to.
A tap is used when there isn't a pipe, like when you tap a keg of beer, or tap a maple tree for syrup. Tap is a little unusual because it can also be the act of tapping as well as the device you use to control the flow. A tap has the sense to me of poking a hole in something that has liquid in it, and being able to keep the liquid from just gushing out with some sort of device. Faucets and spigots are plumbing, with connectors and pipes.
Now I know for a fact that in other areas of the US, folks use "tap" or "spigot" the way I use "faucet". We can still understand each other, so it's fine to use them as synonyms.
As mentioned in the comments, in British English, "tap" is used for all three, which makes sense because "tap" is from Old English and "faucet" and "spigot" are from Old French according to an online etymology dictionary.
