To open a disk image, you just double click on it. The image mounts as a new volume on your desk top. This usually looks like a white hard drive. You can then double click this virtual drive itself to open its contents in a new Finder window. Copy items from the disk image by dragging them to your desktop or to your folders.
When you're done using the disk image, you can eject it. Either drag it to your trash can or Control-click (right-click) the volume and choose Eject from the contextual pop-up menu. When you reboot your computer, the disk image automatically unmounts. So if you wonder "where did that folder with my Application disappear to?" it may be that the disk image unmounted.
That brings me to the most important thing about disk images. You're not meant to run applications from them. When you download software that uses disk images, the idea is that you'll mount the image, copy the application to your hard drive (typically to the /Applications folder) and then dismount and discard the disk image. This is why so many disk images include an alias to your Applications folder. They're trying to make it easy for you to install the software. Just drag the new program onto the Applications folder--either the real one on your hard drive or the virtual one on the disk image. This copies the software into the right place and you're good to go. Eject the disk image and launch the copy in your /Applications folder.