You can find Normal Items everywhere throughout all difficulty levels of the game. Also, you can find them by gambling or at town merchants.
Exceptional Armor and Weapons
Exceptional Items have their Character Level requirement around level 25. There are Magic, Rare, Set, Unique, Socketed and Crafted versions of these items. They appear in the later Normal difficulty levels and they become more frequent at higher difficulties. You can find Magic Exceptional Items at the Town Vendors or and Rare Exceptional items through gambling.
Exceptional Armor Items provide improved Defense but they require more points in the Strength attribute.
Regarding Exceptional Weapons, they provide a higher Minimum and Maximum Damage, but they require more points of Dexterity and Strength. In addition, these type of weapons provide more +skill points, and higher level spells and skills.
Elite Armor and Weapons
Elite Items have their character limits at 39-73 levels. There also Crafted, Socketed, Magic, Rare, Set, and Unique versions of these items. Firstly, they appear at a later stage in Nightmare difficulty level and then more frequently as you level up. You can find Magic Elite Items at the merchants and Rare Elite Items by gambling.
Elite Armor Items offer increased Defense with a cost of higher Strength requirement.
Elite Weapons provide better Average Damage but they require more points of Dexterity and Strength. The Weapons provide higher level spells and skills and more skill points as well.