What is it?

[Daddy-o]

HT is this coral? Any difference in the orange and grey portion? Green in middle is the mouth?

And of course, how do I care for it?

 

[Flreefer]

That is a Ricordia florida mushroom coral.
Yes, the green center is the mouth. It doesn't require any special feeding, it gets all it needs from light, and what it can pull from the water.
It just needs low-med light and low-med flow.