Tamsinblevin - I can't say that I've ever heard them "off-board diagnostics" before. Anyway..."on-board diagnostics" refers to the computer's testing and monitoring of various systems, sensors, and actuators. When it detects a fault, it alerts the driver through a set of warning lights. The most common would be the dreaded check engine light. Cadillac often uses two lights, service engine soon and service engine now, depending on the seriousness and the internal prioritizing of the detected fault(s). Other examples of on-board diagnostics include ABS and air bag lights. Basically, the computer systems are testing themselves and warning the driver of a problem resulting in possible increased emissions, reduced performance, or a malfunction in the vehicle safety systems.
From what you're asking, "off-board diagnostics" seems like what we refer to as "no code diagnostics". The engine is just not running right, but the computer does not detect a fault. A perfect example of this can be found on Ford's 4.6l V-8's. The throttle body has a tendency to get dirty, resulting in a rough idle, hard starting, and stalling at times. The computer will rarely determine a problem. No-code diagnostics can be time consuming and aggravating. When the computer can supply a trouble code, it may not tell you the exact problem, but it's usually a good place to start. Without a code indicating the proximate cause, it becomes a matter of checking the basic engine functions which aren't generally monitored directly, i.e. compression, fuel pressure, etc.
