For example, I'm watching CNN. Wolf Blitzer is speaking on the background, there's a news ticker feed on the bottom and occasionally, individual video clips are shown on the right to go with the story. How is that all put together, live?
What software / hardware is used by TV companies to generate a live video feed that can be made up of smaller, live pieces?
In general, it is done with a production switcher/vision mixer. I believe CNN uses the Sony MVS8000. Specifically, it is a DVE or resizer feature that scales and positions the video. The video is played back from another device, likely a video server, under the control of the newsroom computer system.
They may also be using whatever software that does the real time rendering of the ticker to do the same thing that the production switcher's resizers do. Many broadcasters use renders from VizRT or Orad (now owned by Avid). Whether it is done in the production switcher or in the ticker software is a complex system architecture decision driven by the graphical requirements.