A complete protein is a protein source that has all of the essential amino acids necessary, that your body cannot make on its own. An incomplete protein is lacking in one or more of those amino acids. A complete protein would be meat. You can find complete protein sources in meat, eggs, cheese, fish, and so on. Incomplete sources of protein would be nuts and grains.
I think the important issue for someone when they think about complete versus incomplete protein is not so much for those who don't shy away from animal products, but you need to make sure that if you're someone who's a vegetarian or a vegan that you're mix and matching your proteins so that you're making your protein complete. Your body can do nothing with incomplete proteins all day long. If that's all you eat you're not going to make muscle and tissue. That's why the challenge for vegetarian or vegan bodybuilders is a lot more difficult and harder to put on lean tissue.
So you want to make sure that you match. For instance, you might want to have rice and beans. You might want to have peanut butter on multigrain bread. That's going to help you to complete the profile so that your body's getting the complete protein that it needs.