A high quality protein is a dietary protein source in which a high percentage of the source's protein can be used by the body. The term high quality protein refers to two slightly different things. Foremost is that a high quality protein can be readily broken down in digestion for efficient use in the body. Second, the body requires certain types of protein, or more specifically amino acids, that it can't produce on its own. Generally, high quality proteins become readily available for use and contain the full complement of essential amino acids that the body cannot produce by itself.
Proteins are a wide class of typically complex molecules made up of chains of amino acids. When protein enters the body, it is broken down in the stomach and small intestine to make its amino acids available for forming the proteins of the human body. There are 20 or more amino acids used to form proteins. The body can make, or synthesize, some of these by itself, but there are at least nine that must come from the diet. High quality protein contains these nine "essential" amino acids in forms that can be readily broken down for use.
High quality protein is most associated with animal protein sources like meat, including poultry and fish, dairy products like cheese and milk, and also eggs. Which of these is of the highest quality is a matter for debate, but eggs specifically have long been associated with high quality protein. Some animal sources can come with specific considerations that might limit their overall protein quality. Most grain and vegetable sources are generally incomplete proteins, and may also contain factors that limit the availability of proteins.
Most people in the developed world don't need to be concerned about getting enough high quality protein, because their diets already provide enough. Vegetarians, who usually eat eggs and dairy products, likely needn't be concerned either. There used to be some concern about vegans, or people who eat no animal products at all, and whether they get enough protein in their diet. It was sometimes recommended that vegans eat complementary plant proteins, or plants which together have all the essential amino acids, in the same meal to get the needed amount of protein. It is now mostly recognized that, in the case of vegans, regularly eating a wide variety of plant foods supplies enough protein.