Bear
It is called the arbor.
It is threaded into the frame to the proper position and orientation which is precise to within about two thousandths and less than half a degree. If it is very far off, the pistol will not go back together.
Then it is pinned in place with a round pin. The pin can be found by looking into the hammer channel at the rear of the frame. If the pistol has been fired a lot, you may have to clean up the hammer channel so you can see the back of the arbor and the pin. To get the arbor to turn out, you must remove the pin. This pin is only about 3/32 of an inch in diameter. If you have a steel frame revolver, the pin is likely softer than the frame and so the drill will want to follow the pin. If the revolver has a brass frame the drill will want to wander off of the pin into the brass. You will almost certainly need a drill press to do this job.
Some folks have said that you can remove this pin with an easy out. But by and large (IMHO) there are two problems with this technique. First of all, an easy out is used for threaded fasteners and the pin is not threaded on 1860 and smaller frames. It is an interference fit. Second, I have not seen an easy out small enough to get into this pin. You might be able to thread it for a number four screw, turn a screw into it and use that to pull out the pin.
Another technique might be to completely drill the pin so the entire pin is sacrificed. Then when the arbor is properly positioned, drill and thread the arbor and hole so as to accept a headless screw.
When you reinstall the new arbor, the dimension from the recoil shield to the hole for the wedge is important. This is so the arbor, barrel and cylinder fit together properly. Also the alignment of the arbor with regard to the wedge hole is pretty critical. The centerline of the wedge hole must be coincident with the centerline of the wedge hole in the barrel, or you will have difficulty getting the wedge to go through.
I have done this operation on a brass frame revolver and can't say I am thrilled with the outcome. It was a junk frame so it did not matter The arbor I installed was not the correct one for the revolver so the barrel never did fit.
I might ask why you are changing out the arbor?