Because amino acids are basically the building blocks of tissues, protein has several functions in a dog's body. It's the structural component of connective tissues, as well as hair, skin and nails. But it's also essential for the immune and musculoskeletal systems. Amino acids also provide carbon chains required for making glucose. Protein helps regulate a healthy acid-base balance while comprising enzymes and many hormones in the body.
The average adult dog will need to replace skin and hair constantly, as well as enzymes and other needs. A diet with plenty of protein will help maintain these replacements -- this minimum need for protein is known as a maintenance protein requirement. This also helps explain why growing puppies and pregnant or lactating females need more protein than the average dog -- they need it to grow new tissue or produce milk. Likewise, very active dogs and working dogs need more protein to maintain and build extra muscle.
To complicate the matter a little further, it's important to note that not all protein is created equal. Some protein sources are more digestible than others, and some sources are better at providing all the essential amino acids. Your dog can get by with low-quality protein, but will need to eat more of the food in order to meet its protein needs. The pricier, commercial dog foods usually come with high-quality protein, whereas cheaper dog foods often contain low-quality protein -- something to keep in mind when you're in the dog food aisle.