So what I'm trying to ask is say I want to download Firefox. If I use 'sudo apt-get install firefox' does it get it from the Firefox website or some third-party repository? Or is using 'apt-get install' the same as getting it from the website?
So most of the packages you'll get using
apt will be from the official distributor/creator, excluding a package from a
ppa (personal package archive). Here is some info about apt. As @guiverc commented you can use
apt-cache policy <package> to check where a package has come from, but also the version you have and the version you can update the package to using
apt. If you want to make sure you're installing the latest version possible it's recommended to get the application from the provider's website. There's a vetting process for distributors to push their updates to the official apt repositories.