First time boat owner obviously from this question, but I was attempting to change out a tire on my trailer and when trying to remove the lugnuts, the peice of the wheel that the lugnut screws on to broke off. It was rusted pretty bad.

A. What is the name of the peice that broke?
B. Is this replacable by me or should a professional do it?

The bolt broke off, that the nut screws onto? You can change the whole hub yourself. Just go to Tractor supply or something and get them. Take the hub with you to make sure you get the right size.

Those studs should be able to be driven out the back side with a sledge and punch. Then take the piece to autoparts store and get replacement. The new one will have larger splines on it but will seat when the lug nut is tightened up .

No offense but if you dont know what it is called take to somone, get the hub fixed and have both serviced

I 100% agree with Hop. Take it someone who knows what they are doing and ask questions. Learn from the experience if you can. In the long run, you'll end up ahead. When I ground up restored my trailer, I found out the hard way one of the bearing races were actually part of the hub itself and could not be changed. First time I ever ran in to that. Let people who do that for a living do it and ask if you can watch/learn how it's done. I'm a first time boat owner, but the first mechanic I went to actually let me supply some of the labor, so I could learn how things work and be a safer boater.

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Best advice!