An Adventure Game is a game that focuses on exploration, platforming and combat. It usually involves puzzles. The Legend of Zelda is a good example. As a guideline:

If it has a levelling system, it's not an Adventure. It's an RPG. (Upgrading items is okay)

The line between 'platformer' and 'adventure' is pretty slim, but I'd say that Super Mario 64 is the most adventurish platformer, and Legend of Zelda is a pretty platformish adventure. Use your best judgement: I'll move it if it's in the wrong category.

If it doesn't have a fair bit of combat, it's not an Adventure. It's a Puzzle.

And there you have it!