Nut size (A.K.A. diameter) is the size bolt that the nut is used with. The same is true of the thread count. Thus a 1/4"-20 nut fits a 1/4"-20 bolt.
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A hex nut in smaller sizes for machine screw diameters under 1/4.
A standard six sided nut.
A hex nut with a nylon ring to resist loosening.