Magento – Current password field cannot be empty in Magento

As of at least 1.9.2.x there is a setting Under "Configuration > Customers > Customer Configuration > Password Options" named "Require admin user to change user password"

Flipping it to no does exactly what you'd expect, it removes the "Admin password" requirement when changing the password of customer accounts. This is presumably to allow for stronger admin passwords using a password manager.

That setting will not work on the passwords of backend users, only customers. That's probably all that most of us need.

It's the current admin password of the user you are logged in with, not the one you are about to create.