Nominal generally refers to pressure treated wood.
4x4 Nominal fits 3.5" x 3.5" Post Size
5x5 Nominal fits a 4.5" x 4.5" Post Size
6x6 Nominal fits 5.5" x 5.5"
8x8 Nominal fits 7.25" x 7.25"
Actual general refers to rough lumber (not surfaced on 4 sides) or vinyl or composite posts.
Some of these post may still measure between 'Nominal' and 'Actual' sizes, particularly in Cedar material.
Diagram illustrates dimensions of a 4x4x48″ arbor. Many other sizes are available.