Both printing proceses can be used in a multitude of applications and sometimes vary in "name" per application or process. Generally speaking....
The left/solid color is called the "flexo" process. Where ink is applied via rubber plate or stamp. A good example of this is a small bag of Fritos where clearly one solid color is applied at a time. Like the chip it is a 2 color flexo process green then black.
The right/halftone image is called "offset 4-color" process where one color is comprised of several colors. Ink is transfered via metal litho plate. Common examples are newspaper, magazine printing etc.. Hope that helps, Eric