When should I use them, and how do I pick one?
(NB: I know the answer, but think this question fills an important gap.)
In printing, a spot colour is an ink that is premixed to the colour required and printed from a dedicated plate, rather than being simulated by overprinting dots of ink from the cyan, magenta, yellow and black plates (4 colour CMYK process).
It may be a colour which cannot be achieved in CMYK, such as metallic or neon. It may be a colour that is achievable in CMYK, but given it is printed as a solid colour rather than overlaid half-tone dots, it will give a better appearance, especially close up and/or on lower-quality stock, where halftone dots need to be printed more coarsely.
You tend to use them in one of two scenarios:
In most of the world (at least UK, Europe, US), the usual way of consistently specifying a spot colour is to use the PANTONE Matching System (PMS). All Adobe software includes PMS palettes. As far as I am aware, free/open source software doesn't, because it's proprietary.
So corporate style guide palettes will often have PMS references for print, along with CMYK values where spot printing isn't feasible, and RGB values for on-screen work.
A PMS reference is usually "PANTONE" (or unofficially "PMS") plus a number, so a red might be specified as "PANTONE 186" or "PMS 186". But they are sometimes a bit different: "PANTONE Process Green", "PANTONE Warm Black 3".
Given you are dealing with ink, and colours that can't necessarily be reproduced even on a calibrated screen, the reliable way of previewing PMS colors is to look at a printed sample book. These are expensive, so unless you need to do this a lot, you'll probably want to borrow one, get your print rep to bring one to a meeting, etc. You can also get tear sheets of single colour samples.
Sample books come in different versions to show the effect of printing on different papers, e.g. if if the sample you're looking at is labelled "PANTONE 186 C" it's on coated paper and if "PANTONE 186 U" it's on uncoated paper. It's important to note that these both represent the same ink.
In software, you may be able to pick "PANTONE 186 CVC" and "PANTONE 186 CVU". These are again the same ink, but refer to on-screen RGB (Computer Video) simulations of the output on coated and uncoated paper respectively.
re: how do I pick one?
My favorite anecdote for this is from many years ago as I was chaperoning a group of design students through the Walker Art Center's in-house design team's offices.
They had us gather as they showed us how they pick colors for their posters, which were often two-color spot printed.
The process was as such:
Obviously not a solution in every case, but I really enjoyed seeing that particular process in action. Sometimes solutions are much simpler than we think. ;)