With windows 8, is it possible to create an application that is always visible? For instance, in previous versions of windows, there is the task bar with quick launch icons. Can I create something similar to the quick launch icons that are always on the screen?
If you are referring to a Windows 8 Store app then the answer is no. You can have a live tile and toast notifications that provides updates to the user which may cause the user to launch your application.
A good article to read to understand how your Windows Store apps will run on Windows 8 go here to learn about Application lifecycle (Windows Store apps). This will explain the App execution state.