Ok then.
Basicly, you can think of Steam as a personal game library. Any game, software or content registered on your account, be it a digital or a physical retail purchase, is available on any PC you use, as long as you have the Steam client installed and you're logged in on your account.
This comes with the downside that anything registered on your account is bound to that specific account, thus it becomes yours forever and can be neither sold, lent or given in as a second hand for discounts.
That said, it doesn't work with games which use a proprietary DRM engine/platform. A game key on Steam, for example, for Valve to offer all the above to you, is unique; there's no one else having the same one and no one else can use it once you did.
Most retail (boxed) games, nowadays, come with one/multiple disc/s containing the bulk of the installation and a Steam key, which will make you register the game on your account and let you download the rest. Since they didn't ask that (you get to download and install Steam during the game's installation process), one can only assume they're not using Steam as their DRM platform; in this case you need to check your games' policies on that on your own, as different games have different mechanisms.
In the end, best case scenario, you would just have to install all of them again as you did the first time and be done with it (which is mainly what happens with games not using Steam or Origin platforms, so you might be in luck); worst case scenario, you would have to buy a new copy of each game :/