cyan (C)  magenta (M)   yellow (Y)    black (K)

These four colours are mixed together using varying percentages to create thousands of colours. When you specify a process colour, you do it by specifying a percentage value for each of the four colours.

This red is 0% C, 100% M, 40 % Y, and 0% K.
This could also be written 0 C, 100 M, 40 Y, 0 K.

A negative is made for each colour: one negative for everything that prints in cyan, one negative for everything that prints in magenta, etc. This adds to the expense of process colour: with spot colours, you'll normally only need two, maybe three negatives. Then add in the cost of a proof, and you'll begin to understand why process colour is often more expensive, unless you find a printer who specialises in colour printing and has greater economies of scale.

You'll need a Pantone swatchbook in order to accurately pick process colours. Keep in mind that not all spot colours translate well to process colours.

What Graphics Should Be Process?
A colour photograph means you'll be using process colour.  If you create a logo that uses four or more spot colours, you may find it's more cost efficient to use process colour than spot colours. Talk to your printer; they can tell you how best to prepare your files.