You almost got it. It's called a "Master Vac". I believe this was short for master cylinder vacuum servo, or something like that. It's more commonly known as a "power brake booster". These can be found at auto parts stores like Autozone. I bought one from them and it costed me like $90 or so. To replace it, you contort yourself under the driver's dash and undo like 4 bolts (nuts?) and pull the pin from the top of the brake pedal arm. Then you go to the engine compartment and unbolt the master cylinder and, being very careful of the brake lines, pull the master cylinder away from the booster. Then disconnect the big vacuum line going to the booster. Now you should be able to yank that sucker off of the firewall and replace it.
As for diagnosing if it's the problem or not, I'm not real sure how. I replaced mine because when I hit the brake pedal, I heard a hissing under the dash and the engine would speed up. Turns out my "master vac" was leaking.
Gene