So the hard drive in your PS4 is what allows you to download and install games. The PS4 initially came with 500 GB of storage. More recent models have been coming with 1 TB (twice the size of 500 GB).
So, originally the only way to get more space was to upgrade what is called your Internal hard drive. Some people were upgrading to 1 TB from the initial 500 GB, and others were upgrading to a whopping 2 TB internal hard drive, allowing people to pretty much never have to worry about running out of space and deleting old installs and whatnot.
Now what this recent update did was provide the option of using an External hard drive to give yourself more space for games. Internal hard drives are literally inside your PS4, so you have to open it up a bit to swap one out and whatnot. But with external, it's a hard drive you can use in the outside simply by connecting it via USB drive and formatting it to match your PlayStation.
What the update did not do was literally provide you with more space, it just allowed for use of a new method of getting more space.
Edit: And then regarding space needed for an install or update, as Chaser said you need more space than the actual size of the download. If the update you need says it is 500 MB, then you need 1 GB of free space. If it is 1 GB, then check your space and make sure you have 2 GB free, etc. Increasing your internal hard drive storage or getting an external hard drive is how you prevent this from happening to make sure you have room for all of your games.
I see. I knew I should had held out on the entire PS4 thing, but no I had to get infamous: second son because of my love for the series. Okay, now I know what I am dealing with and I suppose playing games is out of the question because of space until then.
Why does things happen to me the moment I start having fun? Dx I swear, it's like the universe is against me lol.