What is this part?
 

This is located just behind the drivers headlight. It sprung apart when bumped while changing the headlight...

Bump...

It looks like a pressure hose, but it's hard to follow where it starts/ends from the picture.

I think it is maybe in the evaporative system? It's right between the air filter housing and the radiator support.
It's a diaphragm/switch on a T-fitting. One outlet goes down to what looks like the idle air controller.
One looks like it goes to the rear of the car.
And one goes to T-shaped venturi tube that bypasses the throttle body.

Well, the parts dept had a really hard time identifying it too. Apparently it's just called a 'switch'. I'm pretty sure it has to do with the evap canister in the back of the car.
There's a tiny orifice under the diaphragm and spring visible in the picture, which apparently pulls the diaphragm down when vacuum is high.
The part is >$80, so I don't care what the thing is called, it has JB Weld written all over it...

I found the "switch" you're referring to in the parts diagrams, but I didn't think it was the same part when I was comparing it to your picture. Glad to know it is, granted, expensive part.

I'll know for sure after I fix the switch, but I strongly suspect I've found the source of my "Small Evaporative System Leak" that has been throwing an error code for the last few weeks. As I look at the rim of the diaphragm on this thing, I can see how one side of it doesn't look crimped down upon like the rest of the rim.
I already know it wasn't the gas cap...
So maybe this thing has been bleeding air into the evap system. Come to think of it, this may well be the switch that tells the system that the evap system is leaking. I can't think of any other reason for there to be a vac operated switch between the canister and the manifold.
It would be pretty ironic that the switch they put into the system to check for leaks is the very source of the leak...

Ah, sounds likely. That would make the metal part just below it the pump? I thought is was maybe an idle air regulator.
I've never seen a vac pump on a gasoline car.
Got the switch fixed last weekend. Clamped it back together and built up a thickened epoxy collar around the housing. No more check engine light! Apparently that was the source of my small vac leak.

that is a brake pump check valve it monitors how much pressure is in the booster if to low will activate your brake pump

You are correct, that is exactly what it is. I had to replace mine because the vacuum pump would not stop running. This is used in conjuction with the brake booster to provide vacuum assist reserve if the engine is off.