Not for regular folks, no.

Employers frequently make direct deposits for salaries, and many banks have a bill paying service that can do electronic transfers to pay certain large companies, like, for example, DirecTV, credit card companies, the power company, etc., but not, for example, your local doctor or the exterminator. (Though those services will generally send paper checks to such recipients in the mail.)

But the only electronic transfers that individuals can do between each other is generally known as a Wire Transfer, it's routed through the Federal Reserve Bank, it generally takes like a week, and it's pricey. My current bank charges $6, and that is super cheap. My previous bank charged something like $40 for this service and required that you fill out forms in person at a bank branch.