The CMYK gamut
Print uses a subtractive colour model, meaning that mixing all the colours together produces black, and mixing none of them produces white. This is the opposite of monitors, which use an additive colour model, where mixing all the colours together produces white, and mixing none of them produces black.
CMYK also works with different primary colours than monitors — cyan, magenta and yellow — which means that the colours it can produce are different from the triangle of colours a monitor can produce. The result is that CMYK can produce some colours that monitors cannot, and vice versa.
Since mixing pigment is quite a bit different than mixing light, printing a photo on paper will never look quite the same as displaying it on a monitor. But nobody is printing photos anymore, right?