A few precisions on the use of pure black and rich black...
Some things are misleading and have not been explained in a very technical way; once you understand how things really work, it's easier to make the right choice.
First, black is not gray, it's black. The reason why it may appear "charcoal" on screen it's simply because it hasn't been enriched with the other colors. But it will print black. Inks are semi-transparent too but it's not a good idea to compare a color on screen with a color on paper.
Officially in the print world, black is always 100% K, and a rich black is anything with value added to the other separation CMY.
There's also what's called the dot gain and it's possible you have some settings in your profile that will render it.
As ink is wet and expands into the paper, especially on uncoated papers (more porous), printers need to adjust their curves of colors to remove a certain percentage of the ink to compensate for this; otherwise all the dots on the paper would touch each others and it will create a result that looks more blurry and will also distort the colors.
When you see a black printed on offset, it's not a 100% pure, it has been converted to a 85% black for example, and the same principle applies for all the other separation. But it's still considered a pure black at 100%. Each printer has its own curve for this but they usually have similar values for dot gain depending on the type of machine they use and which stock the design is printed on.
Some extra Stack Exchange related answers here, and information about dots/dpi/ppi/lpi here
When to use a pure 100% black, and not a mix of black
It's better to use a pure 100% black when working with small characters or when the text is printed on a white background.
If a black is enriched for no reason on small characters or graphics, the result might not look as sharp as when using a pure pure because of the misregistration. When press operators calibrate the job at the beginning of the print run, they need to adjust the 4 CMYK plates together and very precisely. That's not an easy adjustment; sometimes they don't really care if it's a low quality print place and sometimes it's their old machines that cannot keep that adjustment for the whole print run. All print shop are not equals, the human factor and investment in good machinery can have a huge impact on the print quality.
That's why on your document when creating crop marks, you will see some little wheels on each sides; these wheels (registration marks) are one tool the press operators use to make sure all the colors are well aligned, that's why they are printed in 100% in each color and very thin. But since machines have some inconsistency, and that process is still done by humans, sometimes the plates are not perfectly aligned; when this happens, you will notice a faint shadow of the other CMY plates appearing around the black, like the sample below.
That's the main reason why it's not always good to enrich the black if it's not necessary; sometimes it's better to use an overprint or a color of black that has very low density in the other CMY so that effect is less visible (eg. 15% or less). And even if the print plates are perfectly adjusted, this can still be a bit visible sometimes.
It's not always the fault of the press operator if this happens; machines are machines, they need to be constantly adjusted too during a long print run, and by default, NO color separation is perfectly aligned on top of each others... otherwise there would be no colors, only a mix of brown! Each CMYK plate has it's own angle, the dots on them technically don't really touch each others.
Some Stack Exchange answer with example and reference.
As you can see, the same problem can happen on the high density colors.
When to use a rich black
You will need to use a rich for a few reasons and to avoid a few issues as well.
First: If you print a pure 100% black-only on top of another color or picture, the background may appear through the black and will be combined to it on the print plate.
One reason for it that some prepress specialists or RIP system have default settings to overprint black that is at 100% black-only; it will create a "multiply" effect and merge all the black together (example below.) For example, on one part of the design, the black will look rich because the other CMY on the picture or color under it will enrich it (saturate it), and the black-only part on the white area or other full color will have a different color value/recipe. In other words, it will overprint that part of your design. on the other parts
It's not always obvious to see this on screen because of the different gamma and calibration of the display, so when using black surfaces or texts that overlap other elements, it's better to add some values in the other CMY colors to make sure this doesn't happen. It doesn't need to be a high density of black; it can be only 1-1-1-100; by doing this, you make sure that a 1% value is added to the other plates CMY, and to tell the RIP "this is not a pure black but a rich black".
The higher the density of all the colors together, the longer the ink will take to dry, that's why you may prefer to not totally use a high density rich black.
No, there's no RGB colors mixed with the CMYK colors.
I amplified the effect on the example below but that's pretty much what will happen once printed. On your screen though, that black part will look totally opaque and will give cover the image properly.
Second, it may make a more balanced and smooth print finish
In offset printing, rolls apply the inks on the plates, and then that plate will stamp the paper. Sometimes these rolls are not well maintained, are old or there's some saturation of ink on them for high density prints. Sometimes it's simply because of the way the design has been done; with large colored areas, the imposition (eg. How the design is placed large print sheets) needs to be done to create some kind of "flow" when the cylinder rolls on the sheets. If there's huge white areas between high density ones, it will create an effect of stamping because the ink roll will apply more ink in some places and none in others. That's another reason to choose a quality printer who will care about these details.
All this will create the same kind of effect as when actually painting a wall using a paint roller and the color will not be applied evenly.
The same happens for any color or 100% Pantone color used on large colord areas.
Basically, if you don't help a bit the press operator by enriching the black, your black could look like this (I use blue on the example to show the "stamp" print clearly):
That's when you want to use a rich black with enough density in each color, but not more than 300 in total of all the CMYK added together. By adding more density to the other colors, you help hide this effect that you cannot really control otherwise; it's a bit like applying 4 coats of paint on a wall instead of 1, the area will be more saturated.
That's another reason why you have to be careful with any 1-color project and using large 100% areas.
Third, sometimes RIPS ignore your overprint settings
Sometimes, your trapping attributes are simply ignored. It's normal they are ignored because few designers have real experience in prepress and will add these settings where they shouldn't or by mistake. So people at print shops will simply remove them and do the trapping themselves.
This is trapping on the image below. Again it's done to make the colors blend properly together. Each color is outlined with a very small border that will be overprinted on top of the other colors OR overprinted to add the background and foreground color together (multiply effect) OR knocked out to make the foreground color totally equal with the background or even remove it (use on white usually).
It requires some analysis for some print jobs because adding trapping might bold a bit some text or shrink it; so it's done according to the darkness of each colors and how they touch each others. For example, a yellow text on black will have a trapping added to it because the yellow will be hidden by the black, but in the case of a dark blue text on yellow, the yellow will go a bit under the blue instead. If it's not well done or no trapping are added at all, there could be a very thin white border visible around some elements.
So if you used an overprint on your small black text to make sure there is no white border around your text once printed, it's not 100% certain that this setting will be respected. The best way to ensure your small texts are overprinted is to simply use the value of the color on the background and add 100% black to it; this will create the same effect as an overprint 100% black-only on a color without the overprint trapping.
Also, when using a light gray made of a different opacity of black, it's better to use a rich black; this will become a rich gray and will hide the small dots on the print. The lighter the color, the smaller and distanced are the dots; they are clearly visible at less than 20% opacity if there's only white under that gray part of the design. That's why when doing a CMYK project, it's better to use a rich black even for the tones of grays.
A note on trapping: Usually, trapping is done on vector elements (vectors and fonts) but not on rasterized elements (pictures or text done in Photoshop). It a good practice to add some trapping yourself on your Photoshop layouts for this reason. I personally use a 1-2 pixels depending on the size of the image. I do it even for web projects, it decreases the anti-aliasing effect as well.
Other Stack Exchange details on why to use real colors and not opacity
Finally, it's better to use a rich black on gradients
When printed, gradients are split into stripe for each % of density/opacity. These steps ARE visible on large gradients, and lamination or varnish will amplify this effect because it will darken a bit the colors.
That's why it's important to enrich the black in this case; it will soften that effect of steps.
More details, here
What recipe of enriched black to use
There's many reasons why one recipe of rich black is better than another, and it depends on your design.
First, maybe you want to tint your black. If you use a higher value of cyan, the black will look a bit colder, and if you use more yellow it will look brownish. For some projects, it would look weird to use a rich black that has a blue hue if the other colors are warmer, for example. And for some projects, it would look weird to have a tinted black of any color, that's when you'll use equal values in all your color, preferably a little bit more blue to make the dark gray-black and not brown-black. It can be a 5% difference only.
There's also very nice special effects that can be created with rich blacks, especially when using gradients or different tones of this black.
Adding other colors in your black in certain proprotions, will also cancel each others or appear more like a metal black, with no tint. The best way to see how it will look like is to use it on a gradient.
If a picture on your design already has a high value in black, it might be a good idea to use the same value so they don't look like 2 different colors. Of course if this value has a higher density than 300% (which is often the case), the picture should also be adjusted using the levels or curves in Photoshop.
Example of a warm rich black used with a green tinted rich black:
Sometimes it's really hard to choose the right tone of black. You can always ask your printer, he has a "favorite" recipe usually that prints perfectly well on his machines. I like to use 40-30-30-100.
Finally, always set your preference to display black accurately
It's important. This way your screen won't trick you on the real appearance of black. Everything should be set to be shown accurately in every publishing software you use.
Details are always "amplified" when printing on offset, and it's better to control as many details as possible to avoid issues. It's very disappointing to realize a details was forgotten once you hold the printed brochure or catalog in your hands!
Source images: Offset print cylinder - xperienceyoursite.com, Minolta logo - danielsprint.com, Misregistration on black - prepressure.com, Trapping - graphicdesign.spokanefalls.edu, Dot gain close up: netdna-ssl.com, Dot Gain printed picture - underwaterphotography.com