The white plastic thing is an adapter.
Many companies have made speakers, base stations, chargers, and other gizmos for the iPod. The older, full-size iPods are larger than the Nanos, so the Nano will not fit into third-party speaker systems, docks, or chargers.
The white plastic thing is the same width as an older, full-size iPod. If you have a third-party speaker system or charger or whatever, you put that white plastic dealie into it, then put the Nano into that.