If  you are interested in marketing credit cards with your brand, big banks have programs for issuing "affinity" and "co-branded" credit cards.  Examples include cards that are branded by charities (e.g., Nature Conservancy), colleges/universities, consumer brands (e.g., MLB baseball teams).  The issuers then pay the affinity groups or brands for the use of the brand and often a fee for each new account.  Larger brands often work with banks to create reward credit cards that offer rewards connected with credit card (e.g. airline credit cards with airline miles).  


Chase Paymentech is Square’s acquirer. So Square pays Chase Paymentech (subsidiary of JP Morgan, who is the de-facto Acquiring Bank) for gateway fees, processing fees, card scheme fees, and interchange fees.

Square's business model is very much like Paypal's business model but probably with lesser risks because Square accepts card-present payments and one would likely see less chargeback risks for unauthorized payments. 

Based on the ATV (average transaction values per merchant segment), the blend of funding (split of credit versus debit card); type of cards (Amex, Visa or MC), you can then elaborate on Square's cost of accepting payments. You can  then add the gateway fees, cost plus small profit that JP Morgan charges to Square and other transaction losses (fraud and merchant default risks) and convert the sum into a percentage (also called basis points). Subtract this basis points to their 2.75% fee and you will arrive at Square's approximate profit or loss (as this does not consider Operating Expenses, etc.).       

Issuing banks make money by the following ways


         Finance charges (customer doesn't pay down to $0, they get charged x% APR).
Merchant revenue (a % of every transaction from a merchant and/or a fee).
Annual fees and penalty fees (late fees, non sufficient funds fees, etc).

There are a few other revenue drivers worth understanding that are especially relevant for banks interested in scaling their business or start-ups hoping to attach themselves in some way to a bank's business model:

Fees from certain transactions such as balance transfers or cash advances.  These are pretty large revenue drivers, especially for newer accounts where balance transfers are most common.  Fees run 3%-5% of the transaction at most top banks.
Opt-in fee products such as credit monitoring or identity theft protection.
Revenue from 3rd parties for qualified leads (customer doesn't qualify for X card, but does qualify for a partner's card, customer gets sent to partner for a finder's fee).
  

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