Proctor & Gamble recently did a survey on consumer thinking about their oral health and one of the statistics was that 33% of the people in the United States believe that a little gum bleeding as a result of brushing their teeth is normal.
It's not! It may be typical, as 50% of the population over 35 has some level of gum disease, but it's not normal. If your hands bled a little bit after you washed them, would you think that was normal? Your gums should never bleed, not when you're brushing, not when you're flossing, or not when you're eating. Bleeding gums means that you have allowed some level of decay to occur in your mouth.
If there is infection around or below the gum line, gradually your gum is going to recede from the tooth, exposing the softer area below the tooth enamel. The result most often is you will end up needing a root canal, or possibly have to have the tooth extracted.
If you have a dentist, you should tell him or her that your teeth bleed when you brush or floss. Usually this means there will be much more bleeding when you get your teeth cleaned, so they can usually see the problem anyway. The hygienist is also going to check how much your gums are receding, and if there are any pockets of infection below the gum line. A dentist can fix this. But you should also be finding out the proper way to brush your teeth--you may be brushing way too hard or with the wrong type of brush if there is bleeding.
If you don't have a dentist and your gums bleed at any time for any reason, this is a clear indicator you need to see one quick before the problem gets any worse. Find a dentist that you like and then you will go as often as you should. Unless you think it's normal not to have all your teeth, then don't think it's normal to have gums occasionally bleed.