Make payments without using third-party gateways

I want to process electronic payments. I am currently using PayPal, but it is too expensive. Many companies can provide electronic payments without the use of gateways. I assume that they directly use the web service of banks.

How it works? Should the payee open an account with this particular bank ?. For example, how does "Pay with Square" work? Does he make a call in different banks?

Is there any documentation?

thank