"Natural" clay sometimes means earthen or pottery/ceramic clay (often gray or brown).
In that case it will dry to harden (over 24 hrs or so depending on thickness), but will be pretty brittle if stressed or dropped unless it's later also fired in a high-temperature kiln--that will make it stronger but the item will still be breakable if dropped/etc.
Other air-dry clays are based mainly on paper/wood or grains as well as various other fillers some of which are even earthen clays, etc.
They will air-dry too and be pretty strong if dropped/etc (depending on shape, etc) but can never be put into a kiln since they'd just burn up. They can be put into a very low-temp oven just to speed up their drying though. There are various qualities of air-dry clay however, with different characteristics and strengths.
(Other kinds of "clay" you might see are polymer clay which is a plastic, never "dries" but hardens with heat in a home oven, and plasticine-type clay which also won't dry but also can't take heat and will never harden.
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