The base to ogive measurement of a loaded round is the distance from the case head to the first point on the bullet from the tip that is full diameter. Different comparators might be very slightly different but shouldn't be off by much. A comparator is nothing more than a machined block of metal with a tight bore diameter hole in it. The ogive of a bullet is in truth the entire tapered portion of the bullet to the tip, but for this measurement we are talking about a specific point on the bullet where the bullet reaches full diameter and is wide enough to contact the rifling. We care about the base to ogive measurement because the most consistent accuracy results are achieved when the base to ogive measurement of your loaded rounds are the same. You may think this would be easily accomplished by simply loading all the rounds to the same overall length, case head to bullet tip. That would be nice, but unfortunately due to the jacket drawing and swaging process of forming bullet there can be a surprising amount of variance even among match bullets in the distance from the tip of the bullet to the ogive point described above. As long as the round fits in the magazine the rifle doesn't care where the tip of the bullet is, so we measure to where it does care about-the first part of the bullet to touch the rifling.
The base to ogive measurement is much more meaningful though if you know the distance from the bolt face to the lands on your rifle. There are various ways to measure this you can research here or elsewhere, but once you have this measurement you now have some information to work with. You can confidently create ammo that will consistently place the bullet the exact distance you want from the lands. In this way you can experiment with seating depth and fine tune a load.
The headspace gauge is a little different concept. Cases of course stretch to fill the chamber when they are fired and need to be resized down a little when we reload them so that they regain a little clearance so we avoid problems trying to chamber the round. We want to resize them just enough that they fit cleanly. The sizing die will squeeze the case down, pushing the shoulder of the case back towards the case head. For a bolt gun we only want to shrink the length of case .001 to .002, semi-autos .003-.005
The question is how do we measure how much we are sizing the case? The answer is that since the entire shoulder is getting pushed back at the same time in the die, it doesn't really matter where on the shoulder we measure to as long as it is the same point every time. You just need something to attach to your caliper with a hole large enough for the neck of your cartridge to fit in so that it is resting somewhere on the shoulder. We don't even care what this length is, we only care about the difference in the length before sizing to after sizing. Something like a 40SW case can be used as a headspace comparator for a 308 because the mouth of a 40 case indexes nicely on the shoulder of 308 brass. As long as you used the same 40 case for the before sizing and after sizing measurement you will be able to find out how much your sizing die is pushing back the shoulder and adjust it accordingly. The nice commercially produced gauges for this purpose are easier to use but the concept is the same.
So to summarize, the case head to ogive measurement is used to create ammo that precisely positions a bullet a consistent distance from land contact in the throat of the rifle. The headspace gauge is used to compare the length of a just fired case from the case head to somewhere along the shoulder so that we can then size the case, take the measurement again, and so determine how much we have pushed that point (and every other point) on the shoulder back.