Answer:
If you saw a man run from a convenience store with a few pilfered products under his arm, he is violating public law. He committed the crime of theft, and that affects everyone.
On the other hand, if your neighbor filed suit against you because your barbeque smoke traveled to his yard, you may be violating private law. You infringed on your neighbor's right to peaceful enjoyment of his property.
Essentially, the difference between public law and private law is whether the act or acts affect society as a whole or is an issue between two or more people.