Normal is the lowest possible grade of quality; it means an item boasts no special purposes, affixes or prefixes, except its own Defense (for Armor) or Damage (for weapons). Usually these items are worn by players at very early levels, when they are yet to obtain even the simplest of Magic Items. Generally, once that is no longer an issue, a player will not wear a Normal item again.
Normal Items first appeared in the original Diablo. This is the most common drop quality, and at many occasions, their damage and defense allow players to use such items long enough, until they can obtain higher quality equivalents.
In Diablo II, Exceptional Items and Elite Items are also considered Normal in terms of item quality, but are higher tiers of the normal items, obtained at higher levels, with same appearance, but altered names and empowered attributes. This confusion occurs due to both categories (item tier, Normal, and item quality, Common) sharing the same name.
Unlike those in the first game, Normal Items now may have prefixes of sorts, that add no new properties, but may slightly increase damage, defense or durability. Sometimes they even may have special prefixes that actually reduce their (already not superior) attributes. Note that Socketed Items appear with gray color, but are still technically Normal.
While there is no real use for Normal items directly, they are often farmed for by players who seek to make use of Rune Words, as those require either a Socketed Item, or a Normal item that can be converted to a Socketed via the use of Horadric Cube.
Diablo III Edit
In Diablo III, Normal items return in two forms: both in the same form as in Diablo II, including the optional minor prefixes, and as so-called Poor (Gray) items, that are also considered Normal for the in-game purposes, but are generally viewed as a separate item tier by players. Poor items are characterized by gray color, and have reduced attributes.
The only real use for Poor / Normal Items in game is to salvage them, obtaining a bunch of Reusable Parts in process. Equipping them is pointless after players get their hands on Magic Items, and selling them to vendors awards too little gold, making most players just ignore the gray items and not even bother to pick those up.