I'm completely at a loss as to what the point of the Viewer Node is. To me, it doesn't appear to do anything. I've read that it allows you to see the results of each node in the tree, but whenever I've attached it to another node, it doesn't change. I don't see any difference anywhere. Some screenshots show it with an image, though I've never had an image box appear on it. What am I missing here?
You either need to have the backdrop enabled or in the UV/Image Editor set to 'Viewer Node'
Form the manual:
The Viewer node is a temporary, in-process viewer. It could be plug in anywhere to inspect an image or value map in your node tree.
Select a view node with LMB to switch between multiple view nodes. It is possible to automatically plug a Viewer node to any other node by pressing Shift-Ctrl-LMB on it.
Using the UV/Image Editor
The viewer node allows results to be displayed in the UV/Image Editor. The image is facilitated in the header by selecting Viewer Node in the linked Image data-block menu. The UV/Image Editor will display the image from the currently selected viewer node.
To save the image being viewed, use Image ‣ Save As Image, F3 to save the image in a file.
The UV/Image Editor also has three additional options in its header to view Images with or without Alpha, or to view the Alpha or Z itself. Holding LMB in the Image display allows you to sample the values.