In the "old" way, each part that could exist on it's own in the real world would be modeled in its own workspace. Then all of the component parts would be assembled in an "assembly" workspace that would let you create constraints (joints) between each component, but no modeling could be done in that workspace. This was a HUGE improvement over the "even older" way. 
If I remember correctly, Creo Elements works in this "old" way and is a free download if you want to see what it's like. You could probably also get the idea from their tutorial videos.
What's been explained so far only tells part of the story. I've worked as a mechatronics engineer for almost 30 years 25+ of tose with CAD and other software. I learend mechanical design and PCB layout on a drawing board.
What has been explained so far is correct in that you don't have to necessarily deal with files as Fusion 360 does not make a distinction between a component and an assembly files, but to a degree that can also be done with other CAD platforms. You can create internal components in Solid Works that are not saved as a separate file.
This is only really another option, however, because at some point in time when you have to break dowen a desing into differnt pieces because the size of a project, at some point in time you will have to deal with different files.
The "old" way is that you don't even have access to a tool such as Fusion 360 because the license was too expensive. Fusion 360 is basically free and even the Ultimate version is affordable. Previously the options in getting a "real" CAD system - not just a moding software or maker toy - at affordable prices were very slim and and only availabel if you were an educatior or still in school and could prove it. Everyone else had to pay full price.
In terms of features Fusion 360 allows you to free-form model organic shapes with T-Splines, which "feels" very similar to modeing with subdivision surface modelers such as Maya or Blender. Then, however you don't have to export that surface from your "mechanoical design" software to add mechanical features such as ribs, studs, threads etc. you just leave the Sculpt environment and go to the Model environment and can seemlessly transition into solid modeling world. that was previously - the old way - not possible.
The other paradigm shif this the cloud based approach which really makes it easy to collaborate with people on a project. It also provides a means to do version and revision management (when Branching and Merging will come). As a cutomer and user of the software ytou don;t have to thing where to locat the server, you don't have to configure and maintain it as all of that works in the background which is great.
Also, if you're a mac user and Engineer, and need a mechanical CAD system there Fusion 360 really does not have any competition.
Modeling in T-Splies feels similar to modeling with subdivision surfaces, but mathematically they are very different.
Subdivision surfaces have a finite resolution. You'll always be able to zoom in to a level where you can disccern between discrete quad surface patches.
That's not the case with T-Splines. Both have their pros's and cons.
Wow! Great posts. Thanks everyone. I think I get it, now! 
I gave kudo to just about everyone too! Thanks, again for all the help. 
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