To enable user of SharePoint 2010 to open PDFs documents without having to save to local machine and then open you need to make a change to the allowed MIME types by adding the pdf extension to the list.
Step 1: On the SharePoint server open SharePoint 2010 Management Shell
Step 2: Run the following command:
$webapp = Get-SPWebApplication http://WEB-APPLICATION-NAME-HERE
$webapp.AllowedInlineDownloadedMimeTypes.Add("application/pdf")
$webapp.Update()
Step 3: Restart IIS