The credit/debit card number is referred to as a PAN, or Primary Account Number. The first six digits of the PAN are taken from the IIN, or Issuer Identification Number, belonging to the issuing bank (IINs were previously known as BIN — Bank Identification Numbers — so you may see references to that terminology in some documents). These six digits are subject to an international standard, ISO/IEC 7812, and can be used to determine the type of card from the number.
Unfortunately the actual ISO/IEC 7812 database is not publicly available, however, there are unofficial lists, both commercial and free, including on Wikipedia.
Anyway, to detect the type from the number, you can use a regular expression like the ones below: Credit for original expressions
Visa: ^4[0-9]{6,}$
Visa card numbers start with a 4.
MasterCard: ^5[1-5][0-9]{5,}|222[1-9][0-9]{3,}|22[3-9][0-9]{4,}|2[3-6][0-9]{5,}|27[01][0-9]{4,}|2720[0-9]{3,}$
Before 2016, MasterCard numbers start with the numbers 51 through 55, but this will only detect MasterCard credit cards; there are other cards issued using the MasterCard system that do not fall into this IIN range. In 2016, they will add numbers in the range (222100-272099).
American Express: ^3[47][0-9]{5,}$
American Express card numbers start with 34 or 37.
Diners Club: ^3(?:0[0-5]|[68][0-9])[0-9]{4,}$
Diners Club card numbers begin with 300 through 305, 36 or 38. There are Diners Club cards that begin with 5 and have 16 digits. These are a joint venture between Diners Club and MasterCard and should be processed like a MasterCard.
Discover: ^6(?:011|5[0-9]{2})[0-9]{3,}$
Discover card numbers begin with 6011 or 65.
JCB: ^(?:2131|1800|35[0-9]{3})[0-9]{3,}$
JCB cards begin with 2131, 1800 or 35.
Unfortunately, there are a number of card types processed with the MasterCard system that do not live in MasterCard’s IIN range (numbers starting 51...55); the most important case is that of Maestro cards, many of which have been issued from other banks’ IIN ranges and so are located all over the number space. As a result, it may be best to assume that any card that is not of some other type you accept must be a MasterCard.
Important: card numbers do vary in length; for instance, Visa has in the past issued cards with 13 digit PANs and cards with 16 digit PANs. Visa’s documentation currently indicates that it may issue or may have issued numbers with between 12 and 19 digits. Therefore, you should not check the length of the card number, other than to verify that it has at least 7 digits (for a complete IIN plus one check digit, which should match the value predicted by the Luhn algorithm).
One further hint: before processing a cardholder PAN, strip any whitespace and punctuation characters from the input. Why? Because it’s typically much easier to enter the digits in groups, similar to how they’re displayed on the front of an actual credit card, i.e.
4444 4444 4444 4444
is much easier to enter correctly than
4444444444444444
There’s really no benefit in chastising the user because they’ve entered characters you don't expect here.
This also implies making sure that your entry fields have room for at least 24 characters, otherwise users who enter spaces will run out of room. I’d recommend that you make the field wide enough to display 32 characters and allow up to 64; that gives plenty of headroom for expansion.
Here's an image that gives a little more insight:
UPDATE (2016): Mastercard is to implement new BIN ranges starting Ach Payment.

Best Solution
This is the LibraryThing founder. We have nothing to offer here, so I hope my comments will not seem self-serving.
First, the comment about Amazon, ASINs and ISBN numbers is wrong in a number of ways. In almost every circumstance where a book has an ISBN, the ASIN and the ISBN are the same. ISBNs are not now 13 digits. Rather, ISBNs can be either 10 or 13. Ten-digit ISBNs can be expressed as 13-digit ones starting with 978, which means every ISBN currently in existence has both a 10- and a 13-digit form. There are all sorts of libraries available for converting between ISBN10 and ISBN13. Basically, you add 978 to the front and recalculate the checksum digit at the end.
ISBN13 was invented because publishers were running out of ISBNs. In the near future, when 979-based ISBN13s start being used, they will not have an ISBN10 equivalent. To my knowledge, there are no published books with 979-based ISBNs, but they are coming soon. Anyway, the long and short of it is that Amazon uses the ISBN10 form for all 978 ISBN10s. In any case, whether or not Amazon uses ten or thirteen-digit ASINs, you can search Amazon by either just fine.
Personally, I wouldn't put ISBN DB at the top of your list. ISBN DB mines from a number of sources, but it's not as comprehensive as Amazon or Google. Rather, I'd look into Amazon—including the various international Amazons—and then the new Google Book Data API and, after that, the OpenLibrary API. For non-English books, there are other options, like Ozone for Russian books.
If you care about the highest-quality data, or if you have any books published before about 1970, you will want to look into data from libraries, available by Z39.50 protocol and usually in MARC format, or, with a few libraries in Dublin Core, using the SRU/SRW protocol. MARC format is, to a modern programmer, pretty strange stuff. But, once you get it, it's also better data and includes useful fields like the LCCN, DDC, LCC, and LCSH.
LibraryThing runs off a homemade Python library that queries some 680 libraries and converts the many flavors of MARC into Amazon-compatible XML, with extras. We are currently reluctant to release the code, but maybe releasing a service soon.