The bore is the hollow centre of any pipe. Nominal is used in the sense of "in name only" or "putative"-- approximate. Nominal bore is the approximate internal measurement across the diameter of the mouth of a pipe.
Significance
International regulations govern the size of pipes' outward dimensions, ensuring pipe is manufactured in standard sizes for easy fitting. Because pipes vary in material and weight, and because of the way pipes are moulded, internal dimensions of pipes that share the same outer dimension may vary. For this reason, any measurement of that internal dimension can only be "nominal."
Types
Pipes are produced in metric and inch measures. Generally, the two types are incompatible except with special transitional fittings. To complicate things further, the two systems use different terminology. In the inch system, nominal bore is abbreviated as NB; in metric systems it is "diameter nominal" (DN).
Warning
In metric systems, pipes, though always measured in millimetres (mm), may be measured either by nominal bore or by overall external dimension (OD). When ordering pipes, verify that when you say 20mm, both you and the vendor know whether you mean 20mm nominal bore -- the size of the hole -- or 20mm overall.