You've probably already seen and used this by now, since it's prominently displayed on the Xbox Guide. But I thought I'd still explain a little bit about how it works. Private chat (called QuickChat during development) is a one-on-one voice chat system. There are a maximum of five channels on the private chat screen: one game channel (if you're in a game), and four private channels. You select one of the private chat channels to invite someone to chat with you. If you're a social butterfly, you can juggle multiple private chats by using the list to switch between channels. When you select one of the channels, the other channels you have open are placed "on hold," (or, as I like to say, "on ignore"). If someone put you on hold and forgot about you, you can page them by moving to the channel you were talking on and pressing Y (Page Gamer). That'll pop a little notification on their screen, reminding them of your existence.

More details in this article at Xbox.com as well:

http://www.xbox.com/en-US/support/systemuse/xbox360/livefeatures/chat.htm