As John says, you download and install it onto each computer you use it on. You then log into an account on Fusion's server and your work folders are there (as mentioned, only you can see the contents unless you choose to share access). So you can open and work on them from a variety of machines. You can manually force F360 to work in offline mode for up to 30(?) days but obviously your work will not be saved to the server until you next go back to online mode. Importing files of different formats is handled on the server and seems to be pretty powerful, although it failed to import some old Alibre files recently – possibly not surprising, as it doesn't have a significant market presence.
Onshape differs in that your PC, tablet, iPhone etc is only a browser display, with the actual graphics computation etc being done in their workstation farm or whatever they call it. So there is no chance of working offline and of course you can't even access your work without an internet connection. Their hobby user deal is extremely limited and all the interesting add-ons are charged at professional prices. I suspect their business model was fixed some years ago and precludes innovative pricing models like Fusion's.
Fusion doesn't require fancy graphics cards to work, unlike some of the established products. However, it's aimed at professional users, not just hobbyists, so full featured rendering, animation etc are available.
Solidworks (Dassault) is available free to most universities it seems and students can get their own full copy for ~£100 for 12 months. Obviously the aim is to get them weaned on to the product, in the same way that Microsoft offer free Office 365 to all school and university students in the UK and many other countries. The shock would come if / when any of them actually tried to obtain a copy for their own use outside of work or education. I forget the numbers but the purchase cost and obligatory annual "support" fees are way beyond any sensible hobby user's budget, with no alternative available, even with reduced features. That was one key factor that encouraged me to find a sustainable alternative and led me to Fusion 360. Until that point, the more time and effort I expended in SW represented more potential future loss. As established market leaders, Solidworks are onto A Good Thing and won't change until the sun stops rising, so as you would expect, they are an arrogant, expensive PITA to deal with professionally (speaking with some experience).
Murray