That part is called a threaded tailpiece; it is the lower part of a drain/strainer assembly.
It is hard to see from your picture exactly what type of sink the drain assembly is attached to, but it does not appear to be a common conventional one.
The problem you are having is not uncommon, but it is not easy to fix. The threaded tailpiece has broken at the base of the threaded part. In other words: you have a thin metal tube with a very fine thread at the top, which threads into the bottom of the drain/strainer body. The tube has broken just at the bottom of the threads, which means the thin threaded portion of the tube is still inside of the bottom of the drain/strainer body.
This will require replacement of the entire drain/strainer assembly, not just the broken tailpiece.