Basically a values thing that they are dedicated to military personnel and their families. Some of their products are open to everyone (IIRC), but most are limited to military, spouses, and immediate family.
From Wikipedia:
USAA was founded in 1922 by a group of U.S. Army officers to self-insure one another when they were unable to secure auto insurance because of the perception that they, as military officers, were a high-risk group.
Actually, USAA is a bank. At least the bank part of them is a bank. They're also an insurance company, an investment company, and a few other things. But the banking part is a bank.
From their site's front page: USAA means United Services Automobile Association and its insurance, banking, investment and other companies. Banks Member FDIC. https://www.usaa.com
Online banks like Capital One 360 don't charge overdraft fees and have fast check settlement. I can't comment on transaction ordering or anything - also, it might be hard to find an ATM that will do cash deposits.
And not only is there no account fee, but the checking account pays a pitiable amount of interest.