Plastic tailpiece on brass bathtub drain?
It's probably nylon, but it's all fine. Almost every concealed drain uses plastic of one type or another these days.

Those fittings are all compression-actuated. Often the entire trap is also so it can be disassembled for cleaning when needed. Don't use caulk or putty. Tighten them about as tight as you can get them by hand (unless you're a bricklayer).