An ECU is generally the car's stock computer. A piggy-back computer works in conjunction with ECU (hence the term piggy back) to give the ability to intercept/alter various ECU signals typically for timing and fuel adjustments.
Motec is a brand of automotive computer. I'm not sure what the VEC1 and VEC2 are, but my guess is that they would be considered piggy-back computers too.
'Piggyback' chips - like the VEC and DASTEK, don't directly control the
spark advance and injectors, they just alter the sensor feedback that indirectly influences spark advance and injectors. They intercept the intake
air temp sensor, coolant temp sensor, and MAP sensor - and send the PCM an
altered signal that allows the motor to operate at a more favorable performance mode. The MOTEC is essentially a stand alone ignition system that
directly handles spark advance and injector pulse.