Omegle has been around for a while connecting random strangers in text chats.
They have added the video chat option very recently.
Created by a Russian school boy, Chatroulette connects millions of strangers in random video conversations.
The web app itself is pretty bare-bones, with two video panes showing the user & you, a huge pane for text chat and a couple of buttons for controlling the chat.
So you can pick and choose the person to have a random chat with.