It’s downloaded probably because there is not Adobe Reader plug-in installed. In this case, IE (it doesn’t matter which version) doesn’t know how to render it, and it’ll simply download the file (Chrome, for example, has its own embedded PDF renderer).
If you want to try to detect PDF support you could:
!!navigator.mimeTypes["application/pdf"]?.enabledPlugin(now deprecated, possibly supported only in some browsers).navigator.pdfViewerEnabled(live standard, it might change and it’s not currently widely supported).
2021: nowadays the original answer is definitely outdated. Unless you need to support relatively old browsers then you should simply use <object> (eventually with a fallback) and leave it at that.
That said. <iframe> is not best way to display a PDF (do not forget compatibility with mobile browsers, for example Safari). Some browsers will always open that file inside an external application (or in another browser window). Best and most compatible way I found is a little bit tricky but works on all browsers I tried (even pretty outdated):
Keep your <iframe> but do not display a PDF inside it, it’ll be filled with an HTML page that consists of an <object> tag. Create an HTML wrapping page for your PDF, it should look like this:
<html>
<body>
<object data="your_url_to_pdf" type="application/pdf">
<div>No online PDF viewer installed</div>
</object>
</body>
</html>
Of course, you still need the appropriate plug-in installed in the browser. Also, look at this post if you need to support Safari on mobile devices.
Why an HTML page? So you can provide a fallback if PDF viewer isn’t supported. Internal viewer, plain HTML error messages/options, and so on…
It’s tricky to check PDF support so that you may provide an alternate viewer for your customers, take a look at PDF.JS project; it’s pretty good but rendering quality – for desktop browsers – isn’t as good as a native PDF renderer (I didn’t see any difference in mobile browsers because of screen size, I suppose).