I had a Pap smear done today and there was blood on the cotton swab after collecting samples from my cervix. Is blood on the cotton ball serious?

Normal. It's actually very common during a Pap smear for some blood to be found on the swabs. Your cervix and it's respective canal, has very sensitive tissue. Remember the swabs, in reality, scrape cells away from the area to be assessed under a microscope. This "scraping" can disrupt the thin tissue lining and lead to a small amount of bleeding.