This is a very difficult matter to grasp. I am working on a chautauqua on it, but to give you an idea:
What your computer calculates in the background does not correspond necessarily with what you see on your monitor. Some colours simply cannot be reproduced on a monitor because they fall outside the range of what it can display. The same goes for printing with four inks: many colours that you can see on your monitor cannot be printed. These colours are said to be "out of gamut". While much depends on the quality of your monitor and then how well it is calibrated, it cannot expand beyond the capabilities of monitors. A printer can use more than four colours. Not only a dektop one that uses six or more inks (they add PhotoCyan and PhotoMagenta to have a more exact reproduction of what one can see on the monitor), but some offset printers also use more colours. Yet these are exceptions. Most only use CMYK.
As for spot colours: when for example making an invitation or a poster, you can choose your colours in a book like the Pantone one. Then the printer knows how to mix that ink, based on a set of basic colours. Yet most of these colours cannot be reproduced by CMYK, and thus also not by your desktop printer. Also, many of them cannot be exactly reproduced on your monitor.
What Photoshop does, when displaying CMYK is on one hand limiting the RGB colours to those that fall inside the CMYK colour space, and on the other hand, saturating the ones that are close to those your monitor cannot reproduce. Yet, if your monitor isn't well calibrated, you get a totally wrong indea.
Print consists of two different items: photographs and photographic documents are reproduced with the four (or six, or..) colours of the CMYK space , whilst fills and text etc is often printed with so-called spot colours. Pantone has a book in which you can see the differences between the Pantone set and its "equivalent" in CMYK. Shocking...
And then there's the different kinds and hues of paper etc etc etc...
