Edit Dec 22, 2012: Outdated. See here instead.Now, with the USGS Seamless Viewer retired, we have to cope with the
National Map Viewer. It's different, but - in my humble opinion - not really more user friendly than before.
After starting the viewer, focus on the map window, don't bother with the overlay data layers on the left. You won't find "elevation" or "DEM" in there. Zoom in to the area of interest and click "Download Data". In the "Download options" pop-up, use "Click here" to download by current map extent. A new dialog pops up, presenting you an "Elevation" option for the first time. Select it and leave the format at the default "GeoTIFF". Click "Next".
That next dialog will show you the available products. You should find 1 arc sec and 1/3 arc sec. I recommend the "dynamic" type as that will indeed download by map extent. Select 1 or 1/3 arc sec (or 1/9 if available). Click "Next". Your selection will appear in the main window in the side bar on the left. Inspect your selection and click "Checkout". You will now have to enter your email address as you will get the download link by mail. I guess that's to keep demand and traffic down, but it's a cumbersome process. Click "Place order" (no charges). Unfortunately you get no information on the data size and the number of files at this time.
Check your email inbox. You should receive message from USGS.gov shortly with one of more links in it. Click the the link(s) (depending on your email/browser software). A new window will open in your browser. It's only now that your order will be processed. Once that is complete, your download finally becomes available. (These last steps are very similar to the Seamless Viewer procedure.)