What is a piece of code that appears to do something useful while performing a harmful and unexpected function like stealing passwords called?

Trojan horse

Trojan horses are programs that enter a system or network under the guise of another program. A Trojan Horse may be included as an attachment or as part of an installation program. The Trojan Horse could create a back door or replace a valid program during installation. The Trojan Program would then accomplish its mission under the guise of another program. Trojan Horses can be used to compromise the security of your system and they can exist on a system for years before they are detected. Reference: Mike Pastore and Emmett Dulaney, Security+ Study Guide, 2nd Edition, Alameda, Sybex, 2004, p 84