Requirements are prerequisites that a character must fulfill to wear an item. If these requirements are not met, the item cannot be equipped and used. Generally, the more powerful an item is, the higher are its requirements; in addition, some items' requirements are adjusted to reflect general progression of a specific class (such as caster items requiring high Magic, but low Strength).
Subtracting or adding requirements is an attribute which can be found on any piece of item, including gems.
This attribute can either deduct or increase the amount of requirements needed to equip an item by a percentage. The Required Level to use the item, however, can't be reduced.
In Diablo III, the only requirements on items are character class and character level. The latter can be reduced by having the appropriate secondary affix on an item (up to -30 on level 70 (-40 with Ancient Legendary bonus), allowing to equip a level 70 item on a level 40 (30) character).
As of patch 2.0., all legendary items have a minimum required level to drop and equip, but their required level changes depending on that of the player's own level at the time of drop. On very rare occasions, one can find an item which is slightly higher on level. The only equippable level 60(70) items that do not have required level are Hellfire Rings and Hellfire Amulets.
In addition to class-specific items, all classes have a fixed set of regular weapon types they cannot use: for example, Wizards cannot use Polearms. Non-class-specific armor can be equipped by anyone.
Follower items can only be used by Followers, and may never be equipped by player characters.