Because your dog's toenails have live blood vessels in them, cutting them too short can cause him pain and make him bleed. Stay calm -- you can easily stop the bleeding without going to the veterinarian. Your dog may need a few treats to take his mind off of the miscalculation, but once you stop the bleeding and calm him down, he'll be just fine.
What to Do
Talk to your dog in a calm, gentle voice, because he's probably in pain. Stroke his fur to calm him down and give him a couple of small treats. Once he settles down, apply a clotting powder -- you should always have this on hand before trimming your dog's nails. Apply it directly to the bleeding nail to facilitate clotting and stop the bleeding. In a worst-case scenario, if your dog won't allow you to apply the powder or you find that you're all out, the bleeding should naturally stop within a few minutes. If it doesn't, contact your vet right away.