The Card Issuer does the Credit History checking.
The card can be issued by Bank or some other financial institution.
The Central Bank/Regulators in each countries set-up an Credit Bureau that acts as a central store of all the Credit history of an individual with any/every financial institution.
It is mandetory for all Financial institutions to report certain kind of transactions like Credit Card Spends/Defaults, Mortage Due, Auto Loan and some other kind of transactions depending on the country regulations.
The data that these Financial institutions need to submit on a regular basis to Credit Bureau includes the amount of loan, the repayments, defaults, etc.
Any Financial institution that needs to give a card or loan can query this information to find out Credit History.
Normally not all Financial institutions directly connect to Credit Bureau, but there are some Reproting companies that connect to Credit Bureau and provide the relevant information to the requested financial institution.
The Credit Bureau or the Reporting companies is not a guarantor. Its only an information provider.
The Financial Institution must based on the information take action as appropriate, ie give loan / card or refuse or charge higer rate of interest or give lower limits etc.
Initially quite a few Card Companies issued cards directly. Even today Amex issues and services its own card.
Master & Visa on the other hand always use tie-up. The advantages with the tie-up being that one get to service a large area and georgaphies where other already have a significant presence. For example Setting up of the Swipe Machines at Merchant establishment & servicing it is expensive & timeconsuming. Plus Marchant may not be comfortable dealing with one more entity. Here Master & Visa leave it to individual Bank to accquire the Merchant. Same is the case with accquiring Customers. Its easy for a Bank to sell me a credit card if I hold an account, they already have the marketing means for doing it.
The results are very clear with the number of cards issued by Master & Visa far exceeding those of Amex. In many countires Merchants refuse accepting Amex because they don't have a good network and dont get paid easily.
The co-branded card is additional flavour to encourage customers spend on a particular type of card.
Every time a card is swpied say for 100%, the merchant gets only 97.5/- Typically 1.5 goes to the Issuing Bank [as they are out of funds for 30-40 days of credit period]. 0.5 goes to the merchant's bank who installed the swipe machine and paid the merchant. 0.5 goes to
Master/Visa [Card Company].
Take an example of Delta Airlines where tickets are booked by a person using Visa Issued by Bank of America. Bank of America also has put swipe machine in Delta Airlines.
In this case the merchant Bank and the Issuer bank is same and hence they would get 1.5+0.5, and Visa would get 0.5.
In order to leverage this, Delta would come out with a co-branded card with Bank of America for Visa. Of the 2% earned by Bank of America, some would be passed to the customer. The customer if he is a frequrent flier would take this card as he gets discounts, airline is happy because its got a dedicated customer, the bank is happy because by co-branding its got dedicated customer of delta booking tickets using its card, the volume increase makes up for the money & discounting. Same principal applies to other co-branded cards.
Hope this is clear.