The front clip is the portion of a unibody automobile which generally includes everything from the dashboard forward. It is created by removing the glass and the doors from the car, and cutting through both windshield pillars (or "A-pillars") and through the floor of the car just forward of where the front seats mount. It is so called because you must cut (or "clip") it away from the car. On a full-frame car the corresponding section (which only goes back to the firewall or bulkhead and does not include the dashboard) is called the doghouse.
A front clip can usually be purchased with or without an engine, and in America is an extremely popular way to get a JDM swap for a USDM Japanese car, as you get the dashboard, hood, fenders, bumper cover, and motor, as well as suspension components which are generally the same as their American counterparts, but are sometimes superior.