When you download an app it's often delivered as a .dmg file. Double-click to expand it and then either drag the app into the Applications folder or run the installer (it'll be obvious which one when you expand the .dmg file).
You can then eject the window on the desktop (Command+e) once you have copied the app into Applications.
Because of Apple's Gatekeeper security, sometimes you may prevented from expanding the file you downloaded if the developer hasn't had it 'signed' by Apple. To get round this, and assuming you are certain that you have downloaded from a reputable source, either right-click or control click on the file and select 'Open'. The setting that controls this can be found in System Preferences>Security and Privacy>General tab - although it's probably best left at its defaults.