What is complete protein?
It is a protein that contains all 9 essential amino acids.

What do you mean by 9 “essential” amino acids?
They are 9 amino acids that your body cannot produce. They have to come from an external source.

So what if I eat incomplete protein?
You’ll get nervous, dizzy, and suffer from exhaustion. Protein deficiency is fairly rare in humans as our omnivorous diets supply complete protein from meat, fish, eggs, dairy, and a few others.

So, who could suffer from a deficiency in complete protein?
People who are on restricted diets. The restriction could be due to ill health, religious practices, or voluntary restrictions in the form of vegetarianism or veganism. This kind of deficiency is very rare, though, and is easily remedied by eating complete protein for a couple of meals a week.

So how can I get the complete protein – the 9 essential amino acids – if I am eating a restricted diet?
By combining multiple sources of incomplete protein :). Hummus is one example – chickpeas are not complete protein. Sesame seeds (tahini) are not complete protein. But the areas they are lacking overlap – the chickpeas lack the amino acids the sesame seeds contain, and vice versa. Whether you eat hummus with bread or slices veggies, it’s good for you as well as mighty tasty.

Further reading:
Bodyforlife2 (very nice graph)
Livestrong (some examples of sources of complete protein)