Don't worry if you accidentally cut the nail too short, even veterinarians, nurses and groomers do this from time to time. Often the nail can seem to bleed a lot! Don't worry too much though, just like with any other small cut, your pet's blood will clot at the wound site and bleeding will stop with time. It can help to apply a tissue and some pressure or a styptic powder could be a great tool to have on hand to quickly stop bleeding. If your pet will tolerate it, applying a small bandage for an hour or so can help to save your carpet!
A Note About Cats
Although sometimes it is necessary to trim the nails of our feline friends, I recommend that you do not trim the nails of cats that venture outdoors. The reason for this is that outdoor cats rely on their nails to climb up trees and fences when escaping from hazards (like unfriendly dogs!). Cats will generally use scratching posts or trees to keep their nails in check, if you notice that your cat's nails are excessively long it could be a sign of arthritis or other problems that are preventing them from being able to do this themselves.
Trimming your pet's nails can seem like an ordeal for everyone involved, but if you take the time to help your pet get used to having it done it can become a great opportunity to build a stronger relationship with your pet.