While it's true that there are an infinite number of numbers, since it's possible that Discover and Visa (for example) have access to all numbers, they might pick the same numbers randomly for two different cards (one Discover, one Visa, both number 101010101001101 for example.) How do they keep this from happening? Or, does it happen so that's why you must specify "Discover" vrs "Visa" or "Bank A's Visa" vrs "Bank B's Visa"?

Akbar B is correct as far as he went. There are other numbers in the account number which identify the issuing bank and type of card as well.

Some numbers are specific in layout to one card. There is a system to identify the card company. Specifying Visa or Discover is just to make sure the transaction (usually electronic) is routed to the right company for payment.

The way they do that is all Visa cards start with 4, all Mastercards start with 5, all Discover cards start with 6 and all American Express cards start with a 3.

The first 8 numbers are usually bank/region specific. I don't know that it really matters after the first 8.