Clay is a general term including many combinations of one or more clay minerals

Clay minerals

Clay minerals are hydrous aluminium phyllosilicates, sometimes with variable amounts of iron, magnesium, alkali metals, alkaline earths, and other cations. Clays have structures similar to the micas and therefore form flat hexagonal sheets. Clay minerals are common weathering products and low...

 with traces of metal oxides and organic matter. Geologic clay deposits are mostly composed of phyllosilicate minerals containing variable amounts of water trapped in the mineral structure.
Clay minerals are typically formed over long periods of time by the gradual chemical weathering

Weathering

Weathering is the breaking down of rocks, soils and minerals as well as artificial materials through contact with the Earth's atmosphere, biota and waters...

 of rocks, usually silicate-bearing, by low concentrations of carbonic acid

Carbonic acid

Carbonic acid is the inorganic compound with the formula H2CO3 . It is also a name sometimes given to solutions of carbon dioxide in water, because such solutions contain small amounts of H2CO3. Carbonic acid forms two kinds of salts, the carbonates and the bicarbonates...

 and other diluted solvents.