Back up all data.
If Adobe Reader or Acrobat is installed, there should be a setting in its preferences such as Display PDF in Browser. I don't use those applications myself, so I can't be more precise. Deselect that setting, if it's selected. Otherwise do as follows.
Triple-click anywhere in the line of text below on this page to select it, the copy the selected text to the Clipboard by pressing the key combination command-C:
/Library/Internet Plug-ins
In the Finder, select
Go ▹ Go to Folder
from the menu bar, or press the key combination shift-command-G. Paste into the text box that opens (command-V), then press return.
From the folder that opens, remove any items that have "Adobe" or “PDF” in the name. You may be prompted for your login password. Then quit and relaunch Safari, and test.
The "Silverlight" web plugin distributed by Microsoft can also interfere with PDF display in Safari, so you may need to remove it as well, if it's present. The same goes for a plugin called "iGetter," and perhaps others.
If you still have the issue, repeat with this line:
~/Library/Internet Plug-ins
If you don’t like the results of this procedure, restore the items from the backup you made before you started. Relaunch Safari again.