Many companies are able to provide electronic payments without using gateways. I imagine they use banks web service directly.
This is incorrect. Basically all web sites which process credit cards (including integrated services like Square) are using a payment gateway. There is generally no way for an average user to determine which one they use, as the details of the payment gateway are not generally exposed anywhere.
Note that PayPal's Express Checkout web frontend is not a gateway. PayFlow Pro is, though.
The customer's bank is never involved directly at all. A typical credit card transaction involves a number of different parties:
- The customer
- The merchant (e.g, you and your web site)
- The payment gateway (e.g, PayPal w/ PayFlow Pro)
- The payment processor (typically a bank, but not necessarily the customer's bank)
- The credit card issuer (Visa, Mastercard, etc, possibly under contract with the customer's bank)
Of all of these parties, the one which you will communicate with directly is the payment gateway. Everything else is behind the scenes.