Normal in Diablo II can mean one of the following:
Game Difficulty: Normal
When starting your first character in Diablo II, you will automatically have a character starting on 'Normal difficulty setting. This difficulty setting dictates monster hit points, damage, immunities etc. It's mathematically impossible for a starting character to play in any difficulty setting than Normal, even if it was possible. When a character kills Diablo, he can then continue to the Nightmare difficulty, playing the game again. When Diablo is vanquished on Nightmare setting, a last testament on Hell difficulty can start.