Proteins are made up of amino acids. It isn't particular proteins that are necessary in diet, but particular amino acids. For humans, these come primarily from breaking down the proteins in foods we eat.
Essential amino acids are the amino acids that humans cannot synthesize; other amino acids can be synthesized from these, but they do not need to be part of the diet. Not all sources of protein have sufficient quantities of all of the essential amino acids. When people refer to a protein source as "incomplete" they mean that particular source is low on one or more of the essential amino acids.
Meat products are typically "complete" because they contain all of the essential amino acids together. Therefore, if you subsisted on various foods but only got your protein from one animal source, you would be okay.
Some plant products do not have all the essential amino acids in large concentrations. However, if you combine protein sources from different "incomplete" plant sources that together cover these deficiencies, there is no disadvantage compared to a single "complete" source.
You also have a fairly substantial reserve of amino acids in the form of all the proteins in the body, which are constantly broken down and recycled. The daily protein intake is just a tiny fraction of the total amino acid content in the body.
Here is a paper that talks about some of these issues, and addresses some of the myths about balancing proteins. The quick summary is that problems develop if you get your sole protein from a source that is low in a particular essential amino acids over a long time period, but there is no need to ensure every meal contain all the amino acids.
It is possible for people on unusual protein diets, such as vegans, to develop a particular amino acid deficiency. There are many many guides on the internet for protein sources to mix to avoid this problem. A common combination is a legume (beans, peanuts) and a grain (wheat, rice). There are also some vegetable sources that provide complete protein nutrition - soy, for example, mostly works as long as it is not truly the only source.
Young, V. R., & Pellett, P. L. (1994). Plant proteins in relation to human protein and amino acid nutrition. The American journal of clinical nutrition, 59(5), 1203S-1212S.