I seriously hope that you mean the part that bolts up to the exhaust manifold, or the rear cover on the turbine. The compressor housing is the brushed aluminum housing that the line from the air cleaner connects, the center piece of the turbo is the cartridge (which bolts up to the exhaust manifold) and the back half is the turbine housing with the exhaust outlet.
As for the broken bolt with the extractor in it, you have a problem. You may have to take it to be professionally removed. The other one can be removed with another extractor if you heat up the housing with a torch first.