No, Harry, you're not a wizard. Strictly speaking, under Dungeons & Dragons rules, Harry Potter would be classed as a sorcerer with a wild magic specialization. Ailea Merriam-Pigg explains:
Non-dragonborn sorcerers utilize wild magic, which can often leads to strange things happening. Since Harry’s known for accidentally making glass disappear, blowing up his aunt, and, oh, you know, burning someone’s face off with his hands, a wild magic sorcerer is perfect. Harry’s standard for battles is basically holding on until he wins, so his constitution is one of his highest stats.
Merriam-Pigg has composed complete character sheets for Harry, Hermione, Ron, Neville, Draco, and Dumbledore. You can view them all here and, as Merriam-Pigg suggests, play them in your next campaign!
-via Nerd Approved