Whenever you are in the position of picking out a new puppy, either from a breeder, a rescue or just from a litter down the street, most people want to get the smartest and easiest to train puppy they possibly can. While a lot has been made about dog aptitude and temperament testing, there is really no IQ test for dogs, at least not one that has been researched and proven accurate. Some breeds have specific aptitude tests, especially for police, search and rescue, and guide and assistance dogs that can be completed at early ages and before formal training begins.
Dogs, just like humans, are going to have varying degrees of natural abilities and tendencies. These are often considered to be a dog's intelligence, although in most cases dogs are simply going to be better at some tasks and training expectations than others. A lot of this has to do with what the specific dog type has been bred to do. Throughout history hunting dogs have developed a very high level of aptitude with scent discrimination, tracking and retrieving while herding dogs have learned to work with livestock. It would be ridiculous to attempt to turn a Labrador into a herding dog just as it would be to expect a Border Collie to become a water retriever. Each different type of dog is going to be naturally more gifted in one or more areas, but within each there will also be levels of intelligence that factor in.
Intelligence in dogs is often measured in how effectively they work with their owner, or how trainable they are. Trainability typically relates to both the components of nature as well as how much time and attention are spent with the puppy as he or she is developing. Since puppies that are well socialized and have lots of regular, consistent and positive interactions with people are more socialized and friendly, they are also more trainable and less likely to be difficult to work with. Combining both a lot of good, positive, consistent attention as well as training to the dog's natural skills and abilities is going to produce a very well behaved, very impressively trained dog.
Watching how your puppy to be is able to solve problems will give you an idea as to how adaptable and ultimately how much he or she can think out new situations. If you make a new sound, watch how the puppy responds. If he or she stops and looks, sits down to ponder or approaches to get more information the puppy is behaving in a normal way. If the puppy immediately begins barking and goes on the attack, he or she is likely very dominant and may be a challenge to train. A puppy that immediately runs away and doesn't stop or come back is highly timid and submissive and may likewise be a challenge to train. Notice this doesn't test intelligence per say, but rather the dog's ability to interact with something new.
Another test that is often used is to give an older puppy or young dog a challenge is covering them loosely in a towel. Drape a towel over the puppy or dog and see how long it takes them to figure out how to get out from under the towel. Be sure to avoid panicking the puppy but let them try to figure it out on their own, the faster he or she gets out from under the better their problem solving skills are. Don't use this test with younger puppies as they are not coordinated enough to physically move around. Any puppy over about 3 to 4 months old is suitable for this type of a test; just make the towel a reasonable size compared to the size of the dog.
How fast your dog becomes aware of the cues and clues that something he or she likes is about to happen also measures your puppy's intelligence. If he or she learns that your putting on shoes and a coat equals going outside for a walk in a couple of days, they are really observant and smart. If the dog knows that when you walk to the cupboard you are getting a treat for them, that also shows a good understanding of the world around them. These puppies will be good dogs for training using hand signals or teaching them tricks based on things in the world around them.
You can also construct simple mazes and obstacle courses for your puppy to help him or her learn and practice problem solving. These can be easy, low cost and fun and use everyday items from the house. Treats can be placed on the other side of the obstacles or under an object that the puppy then has to find and turn over to get the treat. Just be careful with these types of tests and routines as you can teach your puppy a lot of bad behaviors such as how to open cupboard doors or open closets, something you may not want to teach.
Dogs that have been breed as companions often don't have a really strong natural or instinctive intelligence that may be found in the hunting, guarding or herding dogs. Rather these dogs have a type of intelligence that is known as adaptive intelligence. They are able to adopt behaviors that are in synch with the environment, not just repeat a rote pattern of behavior. These are the dogs that know to bring their leash to you when they need to go outside or learn what the words car, treat, walk or food mean within just a few short repetitions.
These dogs, even as puppies, may seem to be more observant of what is going on about them and less concerned about typical dog behaviors. They may also be able to learn by observing other dogs and pets in the house. Some dog trainers consider the ability to learn by direct observation as one of the most important signs of high levels of intelligence. If there is only one dog in the house you may find your puppy is watching and copying you in different types of behavior, which is a sure sign of a very smart pooch.
Regardless of the breed of dog you have, or his or her natural abilities or adaptive intelligence, it is still a mater of lots of socialization and practice that will make them the best possible companion. Without practice and positive interactions with people even the smartest dog is not going to reach his or her potential.