Monitors emit light, paper reflects light. When emitting light, it turns out that you can use a bigger spectrum of colors, plus you have more variation in intensity. This is only improving as monitors get better, have more contrast, etc.
When creating color for a specific medium, you can take this into account. If you want to create an advertisement for a magazine in Photoshop, you use CMYK. You could of course send in your image with RGB color-space, but the result might be that some colors are out of the spectrum, which would result in loss of quality. In that case, the printer translates the RGB value that cannot be printed into the nearest CMYK value.
OK you think, what does it matter if I do this or if the printer (or the image-editing program of the printer shop) does it?! Sometimes it doesn't matter. But for high quality printing where the graphic designer wants absolute control, it does matter.
Aside from CMYK and RGB, there are color spaces like Adobe RGB. These are more specific. You can use a calibrated monitor. When the image is then send over to another user, with another monitor, in another lighting condition, the other user will see the image like the creator did, if his monitor is calibrated as well. That's the idea at least.
In my experience color handling is something very tricky. I don't calibrate my monitor, I don't use color profiles, except for the standard ones installed by default. If you don't know what you're doing, it can work against you.