Plex is a personal media library that has grown to include other features as well. At a simple level, you need the server portion (an app) and it can run on Windows, Mac or even some NASes. You can then add libraries in Plex for media like TV Shows, Movies, Home Videos, Music, etc. Every library can point to a folder of video files, and Plex manages all the metadata. think of it almost as a personal Netflix. From there, you can use the client apps (which are excellent) on iPhone, iPad, Apple TV, etc. They will basically see a full UI with artwork and metadata for all of your libraries. When you create a use account within Plex, the Plex client can work remotely and see your library at home from across the internet. You could even put an Apple TV in a different location and it will see the library sitting on the computer at home (or wherever you have the Plex Server app installed).
Live TV is a newer feature. If you have an OTA antenna, and one of the supported ATSC tuner devices, you can connect the antenna to the tuner device, and when the tuner is plugged into the network (or on wifi) you can use the Plex Server app to browse for it and Plex will scan for channels and automatically grab all the tv guide data for the channels you have available. I assume then within the Plex client you'll see Live TV in addition to your libraries. There is a DVR feature as well (I haven't looked at this much), but it's where the Live TV feature stemmed from. I guess the DVR also used the tuner + antenna to record shows to your local Plex Server, and now Live TV takes it a step further by allowing you to watch these channels live, right within Plex.
Plex Cloud is a separate feature as well (I believe there's a few different cloud options, I played with them a bit but didn't do much). One allows you to use your cloud storage instead of local storage where your Plex Server App is installed. Then it will actually act like a private Netflix so to speak that's running in the cloud (say on DropBox for example).