By default, when you download software, it'll place the icon on your desktop. Double click the icon and it'll either unzip or mount.
Most Mac software requires a drag and drop to the applications folder and its installed. More complicated software like Parallels, Photoshop, Fusion, etc.. have installers that you run like Windows programs. Basically *.dmg stands for disk image and they're portable volumes that need to be mounted before they can be accessed. Think of them like USB flash cards, as soon as you plug one into your USB slot, the flash cards mount as a drive and the contents can be accessed in a folder. Same thing with *.dmg files, as soon as you double click them, they'll mount on your desktop or in Finder. Double click the mounted icon on your desktop and in the folder you'll see one of 3 things: 1) Program by itself 2) Program and Applications Folder 3) Installer... If you just see the Program, all you have to do is open the Application folder in Finder and drag the Program to it. If you see both the Program and an arrow pointing to the Applications Icon, the developer made it easier for you and all you have to do is drag the program to the folder and your done. And if its an Installer file, just run it like a Windows installation program.
After the *.dmg file has been mounted and your done copying it to the applications folder, you can right-click the desktop icon and eject it, just as you would unplug a USB flash card and your done with the file.
That's pretty much to it.