Source: Yugioh Wikia
Materials are cards used to Special Summon Fusion Monsters, Synchro Monsters, Xyz Monsters, and Link Monsters from the Extra Deck. Each type of monster has a corresponding type of Material: Fusion Materials, Synchro Materials, Xyz Materials, and Link Materials, respectively.
Most Fusion, Synchro, Xyz, and Link Monsters have their required Materials listed in their card text on a designated line above any card effect, cost or condition. However, some Fusion Monsters, such as " " Fusion Monsters (which are Summoned in a special way that does not use Materials) and " " (which has complex Materials), do not list their Materials in this way. Similarly, the Synchro Monsters " " and " " do not list Synchro Materials as they cannot be Synchro Summoned. Strangely, " " does list appropriate Materials despite the fact it cannot be Xyz Summoned other than by the effect of a " " card.
While Materials are usually only used in Fusion Summons, Synchro Summons, Xyz Summons, and Link Summons, not all of these Summons always require Materials, and Materials can be used in Special Summons other than these. For example, " " can perform a Special Summon that is treated as a Fusion Summon without using Materials. Conversely, and the monsters in the " " series both Special Summon Fusion Monsters using Fusion Materials, despite not being Fusion Summons.
Fusion Materials, Synchro Materials, and Link Materials are sent to the Graveyard after being used (by default); Xyz Materials are instead placed under the Xyz Monster until detached (or that monster is no longer on the field). Cards such as " " can be used to retroactively attach Xyz Materials to an Xyz Monster.