Geometry?
 

Now, I have an A in geometry honors but I can't figure this one out.

When we talk about the geometry of bikes, barring components, what's the difference?

What separates my XC Hahanna frame from a "Dirtjumper" Kona Shred? Strength for different stresses? Am I missing the point completely?

It's the angles, lengths, and relationship of all of the tubes.

Well yeah but what is the effect on the bike as a whole?

It'll influence your body position and weight distribution on the bike and will determine how the bikes handle in different situations (climbing, descending, in the air, pedaling, etc.)

Ok that makes sense.

So, for instance, a dirt jumper would have a lower seat to keep the center of gravity down and make it more stable in the air while an XC would be taller for easier pedaling?

Yeh sort of, but its got more to do with clearance. If your dirt jumping you want to be able to move around the bike without the top tube getting in your way. As for XC your right, better pedal efficiency. Just look at road bikes, the top tubes are almost parallel to the ground.

Then there is the different head angles etc. For instance a DH bikes usually has slacker head angle which helps controlling the bike at high speeds, where and XC bike will be different again.