The first digits of your card number denote the card network. The next digits help identify the bank or credit card company (e.g., Bank of America, Chase, Capital One, etc.) that issues the card.
This is how you see the bank number on your card:
Visa - Digits #2 to #6 (ex. 12800)
MasterCard - Digits #2 + #3, #2 + #4 or #2 + #5 (ex. 56 or 23 -- the digits don't necessarily have to be next to each other)
America Express - No bank number. Instead, digits #3 + #4 identifies type of American Express card and currency
Discover - No known method of determining bank number
Unfortunately, there is no way to determine what number corresponds with which specific bank. Furthermore, it is possible that one bank can have multiple bank numbers.