"If you’ve ever had a hangnail or cut your nails too short and got to sensitive skin, you know just how your dog feels when he breaks or cracks a nail." ~Dr. Marty Becker
I don't know about you, but I had such thing happen to me enough times. Or a nail caught on something and torn into the ouchy part. It hurts. Our dogs never had that happen, knock on wood.
What should one do if it does happen? If there is bleeding, of course, first you need to stop the bleeding and clean it well with warm soapy water. It is better for the nail to be gone altogether than when it's cracked or split. The remaining bit of the damaged nail needs to come off. Unless you're comfortable with the idea, it's better to make a trip to a vet clinic. For me, that's what I'm doing just so it can also get numbed up for the procedure. Don't forget such things can get infected too.
Whatever you're going to do, do not do nothing.
You may or may not see any blood, particularly if your dog hurts the nail while you're not around. You might, however, notice a limp. When a dog starts limping, examining the affected leg all the way from the toes is a good way to catch such issues.
Don't forget that overgrown nails are more prone to injury as well as certain health conditions make nail injuries more likely.