I am busy downloading Skyrim on PS4 and it needs to download about 10gb of the total to be playable.
What does this mean? Can I play the full game minus the DLCs and mods until the full download is complete?
A Game can be playable before it has finished downloading the core files needed to start the game. If the download does not run in the background, then you will get a message in the middle of your playthrough saying the rest will need to download. I am not sure if the PS4 downloads in the background(I think it does), but if it does, then you can play and, as long as your internet speed is fast enough, then you should be able to play it smoothly as if it was fully downloaded. I am not sure about DLCs and mods but I assume they need files that might not have been downloaded yet.
Core files downloaded to the PS4 are required to play and files will continue to DL in the background if you're not already tying up resources in multiplayer (most games will block the multiplayer aspect until the game is full downloaded)
Typically the the first estimated hour or so of game play is downloaded before you can embark. However, if you have a slow or weak internet connection you'll 'outpace' the DL and get stuck waiting.
Mods and DLC that are part of a digital package are baked in as if you had purchased a disk, but they're not typically loaded among the core files.
Mods are entirely external and are downloaded from a remote server. These files modify existing files that must be present beforehand, thus the requirement for the core file installation first.