First, this isn't really quite a CMS question(except where maybe you'd be looking for a plugin related to the things I mention below). Roughly speaking your CMS generally doesn't care too much what your content is. It's more responsible for what to do with it, eg. organize into categories, generate listing pages, etc.
The overall reason PDFs don't display on-site is that PDF isn't a web-native file format, like say HTML or the various common image types. While Chrome has a built-in PDF viewer, this generally means that displaying a PDF in-browser requires something like a plugin from Adobe Reader, or Foxit. But this still takes you out of the "web" and into your OS system/browser for the viewing. Even then, the plugin generally has to be configured, by the user, to display within the browser. It's often set to always prompt for download.
FlexPaper is an open-source Scribd-like document viewer that will handle PDFs.
Embedit.in is similar(and handles many other file types), but hosted.
You'll have to determine for yourself how functional they are and whether they'll handle whatever particular documents you're likely to run through them. I haven't used them myself to date. I believe PDFObject is the more mature; there may be others. Once you make a selection, there's nothing keeping you from integrating it into your site templates on output.