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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This item is a semi-standard size and is therefore more expensive.
10-32 is a fine thread, but it is listed under coarse thread because it is very common.
A standard six sided nut.
For adding a machine thread to material that otherwise cannot be tapped.