My puppy has a double set of teeth-  is this normal?

No,  puppies and kittens normally lose their deciduous (or baby teeth) just like we do;  As the adult tooth comes in, the baby tooth normally will resorb making room for the adult tooth.  When this doesn't  occur,   the adult tooth may deviate from it's normal position, producing malocclusion. The resulting double set of teeth overcrowds the dental arch, causing food to become trapped between the teeth, leading to early periodontal disease. A double set of roots may also prevent normal development of the socket resulting in early tooth loss. A retained deciduous tooth should be extracted as soon as an adult tooth is noted in the same area as the baby tooth. If extraction is performed early, the abnormally positioned adult tooth usually moves to it's normal location.