If you accidentally cut your dog's nails, it's best to stay calm and apply a coagulating agent, such as cosmetic powder or even ice, to constrict the blood vessels and numb the puppy's pain. Positive reinforcement and treats can help your dog overcome the fear or anxiety caused by cutting his nails. I ended up finding information to apply pressure to the nail with a clean cloth or cotton ball and then packing the injured nail with cornstarch, flour, or baking soda. In fact, not only do I recommend them to my veterinary clients, but I also carry them in my bag for any emergency due to cutting dogs' nails.
If the piece of nail that was cut too short is too small, once the bleeding has stopped, you can allow your dog to return to normal activities. Dogs with white or light nails tend to have quick visibility, so it's quite obvious where to avoid having their nails cut. Standing and walking on toenails that are too long can change your dog's posture and gait, which can cause pain that could last longer than the temporary bite of a nail cut too short. I cut my dog's nail so that it would stop bleeding immediately after a little pressure, but he keeps kicking it and it starts to bleed.
To prevent your dog from licking his foot once the bleeding has stopped, consider wrapping him up for a few hours, putting a sock over his foot (assuming your dog doesn't try to chew or swallow the sock), or distracting him with attention, toys or treats (as long as he doesn't get too excited). Accidentally trimming a toenail that's too short is easy to do, but thankfully the risk of infection is low in most situations, so there's usually no need to take special precautions to control infections, such as antibiotics or aggressive scrubbing the nail. Fortunately, it's quite rare for an infection to occur from a toenail cut that is too short, but I would still try to prevent your dog from licking his nail if possible. You're cutting your dog's nails and you're accidentally cutting off the fast one (the blood and nerve supply to the dog's toenail) with your nail clippers.
Keep in mind that, while it is true that a Dremel cauterizes when it shortens the nails (so there is no blood), the dog will continue to feel pain if its nail is “too short”. Hopefully, by cutting his nails repeatedly, your dog will begin to learn that he has nothing to fear. However, as an integrative veterinarian with more than 20 years of experience (including teaching countless dog owners how to cut dogs' nails), I can assure you that you can avoid future setbacks by cutting your dog's nails and re-trimming your dog's nails with confidence. The easiest and most effective way to stop dog nail bleeding is with a powder or a cosmetic pencil, which can be purchased at most major pet stores and pharmacies.
It seems that your puppy was not carried away by the experience, but if he seems wary of cutting his nails in the future, be sure to combine holding the paw, holding the hair clipper and then trimming his nails with positive things such as treats, toys or attention.