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Showing a purebred dog in the conformation ring is a way to have a dog judged against the criteria for her breed's standard. Judges will give points to dogs that meet the criteria of the breed standard based on build, gait, markings, and temperament. A competing dog can accumulate points until she becomes a champion. Well-built dogs with outgoing personalities and a winning "ring presence" are most likely to be successful in the show ring.
That answer is very broad and in part incorrect. Judges do not have the ability to "give out points". Points are awarded onto the Winners Dog and Winners Bitch and is based on the number from each of those sex that show in the classes. Classes always start with Dogs (which in dog show terms Dog means male and Bitch means Female). The Dogs will have competitors in each of the following classes: 6-9 months old, 9-12 months old, 12-18 months old, American Bred, Bred-by and Open Classes. The first place winner of each of those classes competes against eachother to win Winners Dog, and is awarded points based on the total number of dogs from all of the classes. Then they start over and the females compete with the same class setup, and a Winners Bitch is selected and received the points. Those two winners are then joined by the "Specials" back in the ring. Specials are both male and female dogs that have obtained their 15 points and 2 majors and have earned their Championship title with AKC. The specials and the two winners are then judged for the selection of Best of Breed, Best of Winners (which is only a competition between the Winners Dog and Winners Bitch) and Best of Opposite Sex (which is the opposite sex to the dog or bitch that takes Best of Breed).
The point system is a little more complicated, but the above statement covers "Showing Purebred Dogs" specifically in AKC.