In the U.S., the Food and Drug Administration handles regulations revolving around pet food with "help" from The Association of American Feed Control Officials.
In relation to Nutrition, the FDA states on this page on their website that:
Any claim that a product is "complete," "balanced," "100% nutritious," or claims of a similarly nature that suggests a product is suitable for sole nourishment when it is not, in fact, nutritionally adequate for such purpose is a potentially unsafe product. For this reason, an AAFCO nutritional adequacy statement is one of the most important aspects of a dog or cat food label. A "complete and balanced" pet food must be substantiated for nutritional adequacy by one of two means.
The first method is for the pet food to contain ingredients formulated to provide levels of nutrients that meet an established profile. Presently, the AAFCO Dog or Cat Food Nutrient Profiles are used.
The Association of American Feed Control Officials (AAFCO) has established profiles for nutrients of pet food. These profiles include can be food in this rather lengthy PDF file which on page 3, lays out what the minimum amount of a nutrient that must be present in a food in order for it to be labeled as one of the profiles. If a manufacturer can provide what levels of nutrients are in their product, they can have it labeled as so if it meets the minimum.
The linked FDA page also states:
The alternative means of substantiating nutritional adequacy is for the product to be tested using the appropriate AAFCO Feeding Trial Protocol(s). This means that the product, or "lead" member of a "family" of products, has been fed to dogs or cats under strict guidelines and found to provide proper nutrition. These products should bear the nutritional adequacy statement "Animal feeding tests using AAFCO procedures substantiate that (name of product) provides complete and balanced nutrition."
Meaning if a pet food manufacturer wants to obtain nutritional adequacy by different means, they can have their product tested in trials. Such products will be labeled as so if they are found to be nutritious and not harmful.
Finally, no matter which method is used for the "Nutritional Adequacy Statement," all labels must also states what stage in life that the animal that is to consume it should be in (i.e. "for growth" or "for maintenance"). A product for "all life stages" meets the more stringent needs:
Regardless of the method used, the nutritional adequacy statement will also state for which life stage(s) the product is suitable, such as "for maintenance," or "for growth." A product intended "for all life stages" meets the more stringent nutritional needs for growth and reproduction.