There are a great many different colours modes available, however the 2 most popular and most important are RGB and CMYK. They are used significantly more than the countless other colour modes as they are the most natural for their respective uses. Both are used widespread and serve their own purposes. As such, it is important to know the differences between them and when to use them to ensure they are properly utilized.
Why Are They Different?
CMYK is the industry standard when it comes to printed materials. It is as subtractive process, meaning the inks subtract from external reflected light to create the desired colour. CMYK stands for Cyan, Magenta, Yellow, and Key; the 4 different inks used in the printing process. The key ink is typically representative of black, but is called key as it is used on the key plate in printing process.

RGB is the colour mode used primarily for all digital platforms; computers, phones, tablets, televisions and many others to create images. RGB is additive, meaning it adds different colours of light to create the colour. RGB stands for Red, Green, and Blue; the 3 primary lights used to create any other colour. Because RGB displays generate their own light, they are ideal for dark environments where it could be difficult to see.

What Can They Do?
As mentioned RGB is used for digital screens whereas CMYK is used for printed materials. They are this way due to the nature of their respective technologies. A digital screen requires light to be produced internally to create its image. To do this it uses colour lights; Red, Green, and Blue. In contrast CMYK is printed onto other materials and uses the light from external sources that bounce off it.
To create white RGB uses all 3 colour at full power, whereas black has them all off. Various combinations are used to produced all other colors. All three of the RGB colours have 256 different levels to create different colours. There are over 16 million possible colours with RGB; 16 777 216 to be exact.

Some sources will claim that RGB only has 216 colours and that is because of what is referred to as web safe colours. Web safe colours are a set of 216 colours that would render the same on all displays. For a long time, although it was possible to produce other colours, these 216 were the only ones used to ensure that it would display correctly. As technology has advanced the need for this has faded as modern monitors. Also, displays can consistently display the full range.
To create black, CMYK can use the key plate alone, but if combined with the other colours different types of black can be made. Pure black, rich black, cool black, warm black, and registration black are just a small handful of the many blacks that are used in CMYK.
Unlike RGB, CMYK is incapable of creating white. The white of CMYK is merely the colour of the paper or material it is being printed on. If you were to print an image onto a coloured sheet of paper, any white in the image would be the colour of the paper; and the image would have a colour cast due the paper being treated as white.
Where RGB has a specified number of colours that can be used, CMYK does not. Theoretically it could produce billions, however in practice this is far from the case. Most printers cannot print more than a couple hundred thousand colours. This is due to the modern technology of printers, as well as limitations on how much ink the material can absorb.

How You Should Use Them?
Due to the lower number of colours available, this must be kept in mind when creating files for print. Using RGB can come out drastically different when printed. Many programs offer the option to convert RGB to CMYK, or even work entirely in CMYK; however, the only true way to see what a printed product will look like is once it has been printed, as these programs are merely simulating CMYK using RGB.
The best option is typically dependant on what you’re working on. If you are making vector artwork, needs to be a very specific colour when printed. Therefore, it is best to work in CMYK so that that colour will be printed exactly as intended. In many other cases however, it is best to work with RGB. Doing so allows you to better visualize what you are working on. Once you’ve completed your work, it can be converted to the exact printer profile required, guaranteeing the most accurate print out.

When an RGB file is sent for printing, it is up to the printer to interpret how it should print those colours. This can lead to unexpected results, whereas when it is properly converted beforehand, you have full control over how it will print. This is important to ensure that the end product is exactly what you are expecting.
A process closely related to CMYK, although rarely used, is CcMmYK which involves using, in addition to the 4 CMYK colours, a light cyan and a light magenta. This is generally only used in situations where very light colour tones are required, as it uses much more ink than regular CMYK.
Pantones are used often in conjunction with CMYK. Pantones are a series of 1 867 unique colours used for printing that all use unique inks. Spot colours, typically for logos are recommended to use Pantones. The design is always the same when printed, regardless of printer or paper type. Although they have CMYK alternatives, they are typically inaccurate or inconsistent. It is the only way to get the colour of a Pantone is to use a Pantone.
Some print shop will have unique requirements, perhaps wanting a RGB file instead of a CMYK file. It is best to do as they know they their equipment best and will likely be able to produce better results themselves.
For a quick recap of the difference between RBG and CMYK, watch this video.