Okay, so I know an H38 ballast is for 100 watt mercury lamps. But what about 100 watt PAR38 mercury lamps? Every single one I have seen says it's for an H44 ballast. So I have two questions:

1. What is the difference between the H38 and H44 ballast?
2. Can the PAR38 be run fine on an H38 ballast when it says H44?

H44 is (as far as i understood) higher current (i guess 1.15A), so on H38 (0.95A) it would be underdriven a bit.
I guess the only consequence is, then lamp would not reach it's designed output and it will take longer to warm up.

What's confusing is the PAR38 lamp says its 100 watts, which would be an H38 ballast, but it calls for an H44, so then is it really a 125 watt lamp?

Well maybe it will be fine on an H38 ballast, the lamp I am looking at has a current rating of .85A.

The "H38" or "H44" represent the load characteristics of the ballast and not directly the lamp wattage, as you might have different wattage lamps, that run on (and are designed for) the same ballast, so all bear the same ballast code. That's why you should observe mainly this code, when matching lamps and ballasts, the lamp power is important only to determine, if the fixture is able to handle that heat...

The ballast is rather a constant current source, so it deliver current into the load, that does not (or only very little) depend on the load voltage. But the power is product Volt*Current*PowerFactor (power factor is ~0.9 for discharge lamps run on low frequency sinewave current HID's are rated for), so if you have two different loads (e.g. two lamps with different arc voltages), their power input would be different even when run on the same ballast (assume 1.15A of the H44).
So if one lamp has 96.6V arc voltage and is run on 1.15A ballast, the lamp will consume 96.6V*1.15A*0.9=100W. So this is 100W H44 lamp (consume 100W and is designed for H44 ballast).
If we take another lamp with arc voltage of 121V and run it on the same ballast, it will consume 121V*1.15A*0.9=125W, so now we have 125W H44 lamp.

But what was said before, mercury lamps (as unsaturated vapor designs) are quite tolerant to slight underdrive, so running 1.15A on a 0.95A ballast would not make any real trouble beside a bit lower output (by ~25%; but this is minor, compare to normal aging by 50..60%).

Thank you for the information.

I imagine that H44 ballasts run 100w mercury lamps at 115v 1.0a while the H38 ballasts run 100w mercury lamps as 130v 0.8a.

I imagine that H44 ballasts run 100w mercury lamps at 115v 1.0a while the H38 ballasts run 100w mercury lamps as 130v 0.8a.

Stating it this way is not correct (beside the numbers being a bit off).
Ballast never runs a lamp on any voltage, but the lamp is, what dictates the current.
In other words:
If you change the ballast and keep the lamp, the lamp voltage wont change, but the lamp current will.
If you change the lamp and keep the ballast, the lamp voltage will change, but the current wont.

So better is to say "H44 lamp runs the systam at 96V, so for 100W is expecting 1.15A, while H38 lamp runs at 115V, so for 100W it is expecting 0.95A"