One example of a nominal number is a jersey number.
The figure above shows a white jersey with the number 24 on it. You may recognize the jersey above as Kobe Bryant's due to the design, but the number 24 on a jersey can appear on any number of other jerseys. The 24 is a nominal number in that anyone could wear a jersey with that number, and it would identify them within their specific team. The 24 doesn't really tell you anything about order (ordinal number) and it's not related to how many people are on the team, so it is not a cardinal number.
Other examples of nominal numbers include a cell phone number, social security number, zip code, passport number, and many more. Though all of these numbers specify some specific thing, the numbers have no real meaning relative to another number of the same kind. Given two passport numbers for example, you wouldn't be able to tell which passport was issued first, where or when it was issued, or anything else without actually having the documents in hand.