If it seems like your puppy is always changing shape, he probably is -- even if he's not technically a puppy anymore. A puppy is officially a dog once he's a year old, but he might not stop growing for a while yet. He'll stop eventually, but the day that it happens depends entirely on his breeding.
Variations by Breed
There's no universal predictor of when your puppy will stop growing, but you can use his breeding as a general guide. For example, a small breed can be practically done growing physically when he's as young as 9 months old. Larger breeds, on the other hand, can take up to two years to reach physical maturation -- if you want to see where your dog falls on the spectrum, consult your veterinarian or an animal nutritionist.
Up and Out
Dogs grow in more than one direction, so don't think that because your puppy is done growing up, he's done filling out. Generally, dogs don't get any taller after 10 months, but their bones and muscles continue to develop and grow stronger for the next year or so. While many breeds' weights plateau when they're around 1 year old, remember that particularly large and muscular breeds will fill out more dramatically.
One reason it's good to predict your dog's adult size is so that you can start feeding him adult food at the appropriate time. Puppy food is very calorie-dense, and feeding it to an adult dog can promote unhealthy weight gain and lead to weight-related health problems. In general, a puppy should switch to dog food around the time that he hits his adult height. This means that smaller dogs who reach adult height earlier in life than bigger dogs should switch to adult food earlier as well.
Even if your puppy quits growing physically, that doesn't mean he's done growing mentally and emotionally. The American Kennel Club considers any dog over the age of 12 months to no longer be a puppy, but of course, young dogs keep up their puppy behavior for longer than that. In general, a dog plateaus and settles down when he is around 2 years old. If your dog grows up without discipline or training, though, he may go on acting like a puppy for much longer, like a child that was never taught manners.