This is one of the wonders of the Mac. Most Mac applications are packaged up in what's called an "application bundle." It's basically a glorified folder that contains all of the code and data for the application. So, instead of having an application installed to C:/Program Files for example, the app is self-contained. These programs don't have to be "installed" as everything they need is already present in that app bundle. Most apps save user preferences and other user data to the ~/Library folder, but the app itself can normally be placed anywhere on your system.
You can run those apps from the Downloads folder, though it's generally better to drag them to the /Applications folder on your Mac. This keeps your apps better organized. And occasionally, some apps are improperly coded to work only from the /Applications folder. It's just best practice to keep everything there, but it's by no means required.
Some Mac apps do have an installer (such as Microsoft Office) as they install files in various places on the machine that are required for the app (such as fonts).