To install most any app on your Mac
What you will download from the internet for most Mac apps will be a .dmg file.
When you open the .dmg, you will see two things - a new icon on the desktop that looks like a hard drive and a window with the application in it.
Some apps, like firefox will have an icon for itself and an icon for the Applications folder within that window. Just drag the firefox icon onto the applications folder within that window and it will then install.
Many apps will not have the Applications folder within that window. For those, you will need to open Finder and move it so that you can see both windows. Then drag the icon for the app from it's window and drop it onto the Applications folder in the finder window. It will then install.
Once you have done this - close the window(s) - right click (or ctrl-click) on the icon that looks like a hard drive and select 'eject XYZ app'.
The app is now in your Applications folder, with access by opening finder and navigating to the Applications folder. This is the one you will want to drag n drop to the dock if you wish.
You can also place an icon for your Applications folder on the dock by drag n drop onto the right hand side of the dock. This will give you right click access to a menu of your applications without the need for opening the finder.
Once you have installed the app - you can delete the .dmg or move it to an appropriate folder to keep you desktop clean.
Was just at the kids house yesterday, he got his Mac almost a year ago and found out he didn't know how to install apps either. If you have done as he did, drag n drop the .dmg to the applications folder - you can delete that.
For those that have the habit of downloading a lot of junk, you may wish to: In my home folder, I have created a folder called Desktop Stuff and another called Screeshots. Have changed my default downloads to be saved to Desktop stuff and naturally screenshots to the screenshots folder. Easy way to keep track of all the junk and maintain an uncluttered desktop.