"complete" protein includes all nine essential amino acids. Animal-based foods (eggs, cheese, meat) tend to have complete proteins.

Info source:

http://www.m.webmd.com/fitness-exercise/rm-quiz-protein-myths-facts?ecd=wnl_day_101512&ctr=wnl-day-101512_ld-stry
A protein is said to be "complete" if it includes:
Amino acids are the building blocks of proteins. There are about 20 different amino acids, and your body needs all of them, but it can only make some -- others have to come from your diet. These are known as "essential" amino acids because it's "essential" that they come from your diet. A "complete" protein includes all nine essential amino acids. Animal-based foods (eggs, cheese, meat) tend to have complete proteins.

"Incomplete" protein sources are low in one or more of the essential amino acids. Beans and tofu are considered good sources of protein but are incomplete protein sources since they contain lower amounts of some essential amino acids.