Flaming is the practice of attacking people on a personal level. While flaming is relatively common on the internet, almost everybody will claim they're opposed to it. They may even flame you for having flamed someone. It's all really confusing, actually.
However, there are some fairly obvious things. Responding to someone's analysis of the deficit crisis by telling them that they probably walk funny or weren't breast fed could easily classify as flaming. Making rude comments about an individual's sexual organs or religion would also probably classify.
Personally, I stick to a simple rule: I say nothing about anybody that I can not back up with facts. I do not comment about people's religions, their eating habits, whether or not they smoke, etc. If I decide to comment about someone's level of intelligence, I make sure I have good examples to support my comment. But even then, I do this very rarely.