CMYK is the color printing process used in the majority of printed materials you come across. It is also called process or 4-color.
CMYK stands for Cyan, Magenta, Yellow, and Black. These four inks are printed together to create full color images. The inks in your inkjet or laser printer are these four colors. You can create almost any color using a combination of these (notable exceptions being any metallic or neon color).
K stands for black, you say? Technically the K stands for Key but nowadays always represents the color black. There are conflicting reasons for this out on the web, but my preferred reasoning is that back in the day, the Key color was printed last to add detail and contrast to the page. And they used whatever dark color they had on hand or was affordable, blue, brown, or black.
You can see different colors because light is being reflected off a surface. The CMYK color model creates new colors by lessening the amount of light reflected off of your (usually) white page. Because the inks are taking away brightness, this process is called subtractive. (Alternatively RGB is additive, because it adds brightness to your black screen.)
Why should I care?
Main takeaway: CMYK is used for 4-color or process printing. But when you upload your images from your phone or digital camera, they start out as RGB. You should convert them to CMYK (using Photoshop) before you send files to your printer. This is probably the easiest thing to do in Photoshop: simply open your image, click Image/Mode and select CMYK Color. Save and you’re done.