Premium dog foods are types of commercial kibble that purport to use higher quality ingredients and provide more a more complete nutritional profile for dogs. The Association of American Feed Control Officials has set minimum standards for any dog food labeled as complete and balanced. Premium (and super-premium) labels are held to those same standards, but not higher ones.
There are a few important quality variables in kibble. Those are order of ingredients, how the protein source (such as fish, poultry or mammal) is processed, other ingredients and additives like preservatives and chemicals.
Order of Ingredients
Ingredients are listed in order of weight before processing (which can be significantly different than weight after processing). The first two or three ingredients make up the bulk of food. Most other ingredients are used for flavor, as a preservative or to fortify the food. If chicken is fifth in the list, the food contains a negligible amount of actual chicken.
Protein Source
Many types of meat and poultry, and some seafood, are used in different formulations of kibble. Whole protein sources are the least processed, and called only by their name ("chicken" or "beef"). Common terms you'll see added to the name of protein sources ingredient list are byproduct and meal.
Meal ("chicken meal" or "lamb meal") is a rendered product made from the flesh or tissue (and may include organs) of an animal. It is cooked once and dried before the final cooking process that makes kibble.
Byproduct ("turkey byproduct") refers to parts of an animal other than main tissues, flesh and organs. For example, chicken byproducts are heads, feet, feathers and entrails.
Byproduct should be avoided. Meal is more processed than meat, but it is also more concentrated protein than whole meat. A good premium dog food will often contain both meat and meal. Food that lists chicken first is not necessarily better than food that lists chicken meal first, and chicken second.
Filler Ingredients
After protein sources, grains - mostly corn, wheat and rice - are the main ingredients in regular dog food. Some premium dog food may minimize or remove grain content entirely, but most does not. Gluten from various vegetable sources is often used to boost the crude protein content of food, but is an inferior source of protein for dogs. Wheat and corn have little nutritional value for dogs. Some vegetables and fruits are good for dogs, but should be secondary to the proteins in any quality food.
Preservatives, Chemicals and Other Additives
High quality dog food typically contains no chemicals, few additives and natural preservatives - mixed tocopherols or citric acid instead of BHA or BHT, for example.
Some additives provide supplemental vitamins, minerals and other important nutrients, such as fish oil, which is a good source of omega 3s. In lower quality dog food, additives often serve to make food look and smell more appealing. Kibble might be colored with chemicals or sprayed with animal fat before packaging.