Originally, Pauline was known as Lady, or Queen of music songs, and the game introduced her as Mario's girlfriend. The name Pauline was given after the then-girlfriend of Nintendo of America's warehouse manager, Don James.
When you select Network Play within the Course World menu, you'll see the option for Multiplayer co-op and Versus, as well as the options to play with friends via the internet or locally with separate systems. Playing with friends allows you to create your own private room to ensure unwanted visitors can't join.