Offset printing is the most common non-computerized technique for printing. Also the way to get the best quality, widest selection of colors and cheapest cost per sheet on large print runs. This is almost always what a "print shop" is capable of doing that you can't do with your laser printer. They also get some 14 colors (I don't know the exact number) to pick from instead of the 4 that other processes will give you. (Yes, in both cases you're mixing -- CMYK (4-color) can't hit all the colors that a professional offset printer can give you, and with offset printing it's also possible to pre-mix the ink to the right color in order to get specific spots the exact right color)

The actual printing involves creating a "plate" (which is possible to generate from a computer) which transfers ink to a blanket cylinder (rubber) that then transfers the ink to actual paper. Multiple colors requires multiple plates (and multiple runs). The intermediate (transfer/blanket/rubber) cylinder is where the "offset" in "offset printing" is from; earlier print processes would go straight from a "plate" cylinder to the actual paper. There is a third cylinder, the "impression" cylinder that is generally involved -- it's on the other side of the blanket cylinder, so that the paper is actually squeezed between two cylinders. However, it's also possible to have another blanket cylinder (with its corresponding plate cylinder) simultaneously printing on the other side, instead of an impression cylinder.

Initial cost is high -- plates must be produced, registration has to be tested (and corrected if slightly off), some paper and ink has to be wasted. However, the cost per sheet can come in under the 14 cents or so minimum that color laser printing will cost you.