First, direct to your question, if you can do 15, but can’t do 16, then the weight is probably OK. But if you can do 20 but stop at 15, you’re not doing much to build muscle. You build muscle not by simply moving weights around but by exhausting the muscle. If you are doing three sets of 15, I’m guessing you need to increase the weights.
There is some conventional wisdom, but as in many sports, every individual is different and there are a bazillion opinions.
One fact that must play into weight training, but I’m not sure how, is that there are two types of muscle fibers, slow twitch and fast twitch. Fast twitch is for power, and slow twitch is for endurance. Different people have different distributions of these two types of fibers, and training can also alter the balance.
Arthur Jones, the guy who invented Nautilus machines, espoused the idea that you should do just one set, and set the weight to where you could do at least 8 but not more than 12 reps in good form (going to “momentary muscle failure”). I later suspected that this approach was pitched by health clubs that figured using their capacity to get 300 members to do one set would make more money that 100 members doing three sets. 
I had a friend who was a successful amateur competitive bodybuilder. She told me to do three sets of successively increasing weight and decreasing reps. The first set was considered a warm-up set of around 12-15 reps, then the third set should be 8-12. I suddenly got past my plateau and increased my weights. I concluded that the “one set” theory is OK but very limited.
Last year I decided to add an extra set. Got past another plateau.
Recently I decided to do three sets but work each set to momentary failure instead of just the last one. Started making progress immediately.
Some people do the sets opposite, figuring they exhaust their muscles on the first heavy set, and will do another set of a lighter weight to squeeze out more reps. I have found, counterintuitively, that even if my muscle feels exhausted from lifting a lighter weight 12-15 times, I can turn around and pump out 8 reps of a much heavier weight. I would guess that it’s because different fibers come into play depending on the load on the muscle. But IANAPhysiologist.
Last, although increased muscle size will increase definition somewhat, definition has more to do with lowering body fat than increasing muscle.