An on-rails game behaves much like a train: while sometimes the player can choose which path he goes down, he cannot deviate from it. Sometimes on-rails games even go so far as to decide when the player moves.
Overview
A feature of video games in which the manner that the player travels is pre-determined on a line, as if on rails. Many arcade shooters have this feature.
Comeback

Dead Space Extraction is a "Guided Experience"
The On-Rails Shooter was assumed to have died with the arcade(with a few exceptions), but recently a the genre has seen somewhat of a renaissance with ports of First Person Shooters to consoles without either the tech or controls to support a full First Person Shooter. Recent examples include Metal Gear Solid Touch for the iPhone and Dead Space Extraction for the Wii.
"Guided Experiences"
The latest trend in marketing On-Rails Shooters is to refer to them as being a "guided experience". While recent On-Rails Shooters are decidedly less arcade-y than their fore fathers, they still have roughly the same style of gameplay. Recent "Guided Experiences" include Dead Space Extraction and House of the Dead: OVERKILL.