Traditionally defined as a fibrous cryptocrystalline variety of Quartz, more recently, it has been shown that much Chalcedony is a mixture of Quartz and Moganite, another Silica mineral.
When it is concentrically banded (often in rather wild patterns) it is called by the subvariety name agate. When it is in flat layers/bands it is called by the subvariety name onyx.
[Note: Many non-banded forms of chalcedony - such as moss agate - are often erroneously called 'agates'. True agate is concentrically banded. Mottled and included chalcedonies are more properly called simply 'chalcedony.' Agatized wood is the name given to fossil wood where the replacement of the wood is by chalcedony, but the banding in this case is due to the wood structure - not concentric deposition of the chalcedony - and the material is chalcedony, not true agate.]