Also, it probably won’t hurt to point out that “downloading” something is different than “installing” it.
Typically, for the Mac, you download an installer package of some sort (it can take on any of numerous guises). That downloaded file can reside on your Desktop or anywhere else, but is useless until you do something with it.
Then, following the instructions, you (or the installer program) will go thru the motions of installing the application.
By default, and as already mentioned by my esteemed Flutherites, the resulting application will be placed in the “Applications” folder where it’s already available to all users of the system (unless you’ve locked them out with a limited account or parental controls).