I recently had a pine cut down in my front yard. It was about 15" in diameter and was cut about 2" above the ground. What is the best treatment I can give the stump to get it to rot out of there quickly? I do not plan to dig or use a tree stump removal machine.

If you really want rot then you should drill a bunch of large holes as deep as you can into it. It will be rotten to below the surrounding ground in about 25 years.

Pine trees are not a deep rooted plant. Their roots go more laterally than vertical. This is the reason that most are uprooted first when high winds arrive. I would trench around the stump, cut the roots and pull it out.

I heard that you can drill a hole in the center, put some charcoal in there and light it. I haven't tried it, but I heard that it works.

Drill holes around the whole stump and let water sit in them constantly. There is a product called Stump Out sold at your local hdwr store for like $5. Let that sit in the watered holes for a few weeks (continue to keep water and the Stump Out in the tree). Then by the end of summer, soak the stump with kerosene or lighter fluid (for a grill) for a few days. After the fuel is absorbed and you light it, the stump will smolder away. No flames to worry about. Just like charcoal.