Cutting a dog’s nails too short and causing it to bleed is a common issue. Here is an article that has information about how to trim your dog’s nails and what to do if it is bleeding that covers this topic:
As mentioned in the above, you can use styptic powder to stop the bleeding in most cases. If you do not have any styptic powder, a bar of soap can be used as well.
In the event that the nail is cut too short to prevent it from bleeding with the above methods, you should see a veterinarian help you bandage it properly and apply any additional therapies that may be needed.