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.
Item Quality: Normal
Normal items are made with grey text, and does not have any magical properties. They are only used early in the game, or in order to make sockets or runewords.