There's a pump that runs when the car is on to pressurize the hydraulics, right?
When bleeding the brakes, should the car be on or off so this pump is on or off?
I'm specifically talking about a Honda '03 if that affects the question.
If what you meant was bleeding the brakes at the calipers to remove air from the system, you should bleed the brakes with the car off. While 'pump' was the wrong word to use, the brake booster runs off the engine vacuum (it's a large diaphragm that multiplies brake force), and this should not be active. You can feel the difference by pumping the brakes a few times with the car off, then pushing with a medium effort on the brake pedal while starting; once the engine starts, the same effort will sink the pedal further and put more force on the hydraulics.
When you bleed brakes, you want to have an assistant push the brake pedal while you crack open the line and then close it BEFORE the pedal hits the floor (or you'll introduce air into the caliper). If the engine is running, not only is this unsafe (front axles can spin, hitting your hands if the front is in the air or moving off the stands if the rear is up), but it drastically changes the pressure at which the fluid will shoot out of the calipers and shrink the time to close the bleeder screw before the pedal hits the floor.
Summary: Bleed brakes (at the calipers) with the engine off.
The only pump that runs would be for the ABS system. If you are trying to bleed that system (ABS) you would need to make the pump run, if you are just bleeding the base brakes you would not need the pump to run.
For example you changed a caliper and you need to bleed the brakes then no need to make the pump run. If however you were changing the ABS modulator you would need to make the pump run as well as take some additional steps in the bleeding procedure.
For clarification based on @Ehryk's comment.
Honda uses an ABS motor to pressurize the accumulator in the ABS system. Factory bleeding procedures for the ABS system and for brake fluid replacement both require the motor to be run at specific points in the bleeding procedure. The can be done by starting the engine and letting it run for a minute or so, or by hooking up a computer to the car and commanding the motor to run.
Here is a link to a video on Youtube explaining the procedure.
More info here although it's for an earlier model car. I think the 2003 has the motor mounted directly to the modulator assembly.
If the engine is running, not only is this unsafe (front axles can spin, hitting your hands if the front is in the air or moving off the stands if the rear is up), but it drastically changes the pressure at which the fluid will shoot out of the calipers and shrink the time to close the bleeder screw before the pedal hits the floor.