Viewing the PDF file in the browser (without download) requires an add-on to the client's browser. Google Chrome, for example, has a built-in PDF viewer and can open files directly, while IE and Firefox require that you install a plug-in (the one that comes with Adobe Reader).
There are two other approaches:
Convert the file to HTML, image, or any other format that can be directly viewed in the browser. This conversion can be on-the-fly using a server-side (written in PHP, Python, ASP, whatever language), or you can just pre-convert all files to a readable one.
The other approach, which is the best, is to use a Flash-based PDF viewer (such as http://flexpaper.devaldi.com/). This is easy, flexible and doesn't require writing server-side code. This approach is used by many Document-sharing sites (e.g. http://www.scribd.com/, http://www.slideshare.net/, http://www.docstoc.com/)