The Amazon Kindle Fire tablet runs on a version of Android so heavily customized that it might be considered a stretch to call it an Android tablet as far as the average user is concerned. The does run Android apps, and Amazon makes it easy by giving it direct access to its own Amazon App Store as well as allowing sideloading of apps from your computer. However, installing apps directly from the main Android Market is verboten unless you go through the process of rooting your Kindle Fire to install it.
Those are the basics, but one thing you may not have realized is that Amazon has taken it a step further: redirecting all requests that would normally go to market.android.com and redirecting them to its own App Store app. The process happens via a file hidden away on the Fire called MarketIntentProxy.apk and it's designed to help Amazon determine which app you're trying to find and give you the option to download it from Amazon instead of Google. That method means that even if you disableSilk cloud proxy server or go through the extra step to install a third party browser like Dolphin, this little file will redirect your requests to Amazon's own store.
It's a move that probably fits in with what most users would actually prefer on the Fire — getting easy access to the app as quickly and easily as possible. It's also possible that Amazon made this decision because Google's website for the Android Market does tend to have some aggressive redirects to get users into the Android Market App when browsed from an Android device. Whatever the case, it's still a little chilling in that a user may simply want to browse http://market.android.com to see what apps are there, as he or she could from a desktop browser, and Amazon has singled out that site for redirection.
So if you want to install Android apps on the Kindle Fire that aren't included in the Amazon App Store, you have two options: hunting down the apk file and sideloading via your computer or rooting the device entirely. If you want to browse the Android Market on the web from a stock Kindle Fire, you're basically out of luck.