Help identifying part

So, I've been working on my car for a bit and I've recently uncovered a valve cover leak. But, that's another post. Anyway, I also noticed a mysterious hose hanging off the back of the intake manifold. I found what I believe to be what it's supposed to connect to via a broken t junction. So, my questions are, what is it's function? What part is that junction and where can I get a replacement? What do I need to know about the hose in order to replace the one that's falling apart?

Thanks!

Oh and pictures: