A molecular phase inverter was a device capable of altering the normal structure of matter, enabling it to pass through normal matter and energy. Such "phased" matter could no longer physically interact with its environment, but could interact with other, similarly-phased matter.
At one point, the Klingon Empire experimented with combining a molecular phase inverter with a cloaking device. Vessels equipped with such an interphase generator would theoretically be able to hide anywhere, even inside a planet, and would be immune to conventional weapons. The Klingons abandoned this research, however, due to several accidents which occurred as a result.
In 2368, the Romulans attempted the same experiment, equipping one of their vessels with an interphase generator. This experiment ended in disaster, however, resulting in a forced chamber explosion in their resonator coil, severely damaging the ship. The USS Enterprise-D answered the Romulans' distress call, and assisted them in making repairs. Upon beaming back to the Enterprise, however, Lieutenant Commander Geordi La Forge and Ensign Ro Laren became contaminated with chronitons leaking from the Romulans' cloaking device, and became phased themselves, unable to interact or communicate with anyone but each other and a phased member of the Romulan crew. La Forge and Ro were eventually de-phased when they were subjected to a high-intensity anyon emission. (TNG: "The Next Phase")