There are a finite number of possible problems. The solution will present itself but only after ruling out the impossible (to paraphrase Sherlock Holmes).
1) Are the PDFs corrupt? Probably not.
2) Is your version of Acrobat or Adobe Reader out of date or corrupt? Acrobat Reader is free - the most current version can be found at http://ninite.com
3) Instead of viewing the PDF in the browser, download it to your Windows desktop. Once downloaded, double-click it open in it in Reader. This takes the browser out of the equation entirely since you are simply opening a local document.
4) Assuming you are using Reader (the free version that does not allow editing/creation of PDFs) then uninstall and reinstall Reader and attempt to view PDFs in browser again.
In addition Ccleaner from http://piriform.com will clear temp files and other junk. Run it (save your browser cookies however) and attempt the process again.
BUT: Your suggestiong #3...Download to Desktop...WORKED.
So: This is an Internet Explorer problem, just as it says it is in the ERROR message. I never had a problem opening a PDF file through the internet before.