inbox has turned in to a verb now when direct / instant messaging is more the norm. In the era of email and on-demand (check your messages from time to time) messaging the inbox was the noun describing where your unread messages are waiting for you to read.
But since now most electronic communication is instant (you get notified immediately about an incoming message or you see the message immediately) inbox can be more of an exception.
Specifically inbox turns in to a verb with wider usage of Facebook where the more regular way of communicating is posting messages on somebody's wall, commenting on them or using the Facebook chat (instant messaging, or IM like in Yahoo IM, AOL IM, etc). So when users want to switch from these more popular methods to sending a message more like an email (send and wait for recipient to read and respond) they can request "inbox me" or notify - "I will inbox you".
Inbox-ing is also one of the methods of going in to a private, one-on-one, conversation where only the sender and recipient (unless others are addressed too) see the messages exchanged. In Facebook all friends, friends of friends or public (depending on privacy settings) can read a conversation carried out by wall posts and comments.
Person 1 on Facebook wall of Person2: Hey what's your cell phone number? I have a new phone and I do not have your number.
Person 2 commenting (responding) to wall post from Person 1: Inbox me, I do not want all my 4,699 Facebook friends to see my cellphone number.