Your horse's pedigree is the geneology of your horse. Who the dam (mother) and sire (father) was, and their dams and sires, and back to their dams and sire. Basically, that is what your AQHA pedigree tells you.
Some pedigrees are extended. This means there is more information put down. When you get your horse's registration papers, you'll see the pedigree on it, with the AQHA number of your horse's ancestors and sometimes the colour they were and the year born.
You can also get a pedigree on your horse with all the accomplishments of you horse's ancestors on it. If they have points. They get points by showing in approved AQHA shows and placing well. The bigger the class, the more points your horse will get if they win, or come second, or third, etc.
When you get your membership, you will be able to access the AQHA site and learn a lot more about classes offered, where they are, and what the point system is! Have fun showing your quarter horse.