The round plastic connector with two connectors coming off the side is a vacuum check valve. It becomes brittle with age and the nipples snap off very easily, as you found out. You can make your own, just connecting up the lines, and it will work, but the original has a one-way valve in it. I would guess that the purpose of the one way valve is for the brake booster to retain built up vacuum.
If you want to replace it, it comes with the brake booster line molded on, you just pull it out of the brake booster grommet and push the new one in.
You can check the operation of the valve by plugging the two side connectors and blowing into it - air should pass one way but not the other.