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.
What is a spot/Pantone colour?
Colours that are created without using any of the four "Colour Process" colour screens or dots are referred to as spot or solid colours. Pantone Inc. [ink?] has a variety of stock colour mixes organized by numbers and names for easy reference that are in wide use. They are called Pantone Colours by many.
Anyone can mix a spot colour of their own and name it anything they wish.
When should I use them
Use a spot colour when you need to reproduce a colour faithfully and cannot depend on the accuracy of the 4C printing process. A logo colour is an example. You'll still have to test how it prints, though, since the colour will change depending on what it's printed on. You wouldn't expect the same solid colour to appear the same on a white paper as it would on a brown kraft paper bag. (That's something called "conform to practice.") Don't proof on white and print on brown.
How do I pick one? (alternate answer)
When choosing a colour, again, "conform to practice."
Conform to practice dictates that you act as/do what the user will do. If you choose a colour to use onscreen, pick the colour from the screen. If you want to choose a colour to print, choose the colour from an actual printed sample.
There are a number of inexperienced designers that like a graphic on the screen, print it, and are disappointed that the colour doesn't look like it did when it was onscreen.
If you know that a poster will be used exclusively under sodium vapour lighting, then take a printed proof to the place where it will be displayed to see how it will look there.
Don't expect to get a true idea about how a colour combination looks in a hardware store, for another example, take the samples to the place where they will be used to see how they look in the actual environment.
Colour combinations change with the amount of light due to our visual apparatus and its foibles and peculiarities. Try to match the quantity as well as the quality of the light.
Artwork will not always be seen under standard or ideal viewing conditions. Try to simulate reality for the best applicable/practical/effective results.
Too, not all of us perceive the same, given the same stimulus. Men, have an incidence of colour perception difficulties of up to 7%. Some say more. Test for those when safety and security depend on perceptual discrimination.
Many different factors affect how we should choose a colour/colour combination.
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. ;)