My dog had puppies a year ago. She stopped nursing 9-10 months ago.

The last few days one of the puppies has been nursing on her for about 10-15 minutes at a time. This is been going of the 3 days.

Is this normal, or is the year old puppy lacking nourishment? The mother also seems to enjoy it.

Some puppies will continue to nurse. It is less usual for the mother to let them at a year old. Try applying Bitter Apple to her nipples. If the puppy is getting most of its calories from a commercial dog chow, chances are it doesn't lack anything in its diet.