Normal in Diablo II can mean some of the following things:
Character mortality
The normal mode of gameplay is that a character respawns in town after they die. This is sometimes referred to as Softcore (an unofficial fan term) to contrast with the officially labelled Hardcore mode, where a character dies once and cannot come back.
Game difficulty
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.
Game version
Normal can refer to the original release of a game, as opposed to the Expansion which changes titles, adds classes and new items.
Item quality
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. More information can be found on the item quality page.