The CMYK process is a method of printing color by using four inks: cyan, magenta, yellow, and black. The vast majority of the world's printed material is produced using the CMYK process.
To print on a CMYK printer you need to make sure your files are in a compatible format. When you print on your home printer it will more than likely print in CMYK rather than RGB. This means that what you print at home may not match what comes off the printer's machine. The spectrum of colors available in CMYK is also smaller than those available in RGB, so some colors are not as easily created and maintained in CMYK.
The colors in the RGB gamut are the colors you will see on your computer monitor. RGB is an additive color model that uses red, green, and blue added together in different ways to produce a broad array of colors. The RGB model is device-dependent which means that different devices detect an RGB value differently. This is due to the variation of devices from manufacturer to manufacturer and the devices ability to display the color, which can diminish over time. It may not always be possible to reproduce what you see on your monitor with the available inks.
In addition to CMYK and RGB printing, pantone colors are available for printing.
The pantone color matching system is largely a standardized color reproduction system. By standardizing the colors, different manufacturers in different locations can all refer to the pantone system to make sure colors match without direct contact with one another
There is a special subset of pantone colors that can be reproduced using CMYK, however most of the pantone system's 1,114 spot colors cannot be simulated with CMYK.
The pantone system also allows for many 'special' colors to be produced such as metallics and fluorescents. While most of the pantone system colors are beyond the printed CMYK gamut.
Pantone colors are described by their allocated number (typically referred to as, for example, 'PMS 152'). PMS colors are almost always used in branding. Pantone colors are used on most offset presses.
Digital presses generally try to match the pantone colors using CMYK or RGB, but may not match the pantone color exactly.
Pantones can be a good guide for what color you are trying to match. CMYK and RGB may not match from machine to machine or from screen to screen, but can have a close match.