A for-profit institution that allows customers to conduct banking business anytime through the Internet is known as an online banking.
Lets go through a few terms to find out why that's the answer.
The answer would not be a credit union because a credit unions's main focus is on allowing members to borrow from pooled deposits at low interest rates, this being mostly done in person.
A commercial bank is an entity that offers services to the general public, both in person and online, but it's sole purpose is not helping them online.
A savings bank is a financial institution that receives savings accounts and pays interest to depositors.
And finally, an online bank is an entity that gives you the ability to manage money online with your mobile device or computer, hence, a for-profit institution that allows customers to conduct banking business anytime through the Internet.
Therefore, the answer is online banking.