I need a device to test my application against but don't own an android device. I found the Kindle fire for $40 which seamed like a great deal to get me started debugging my app but it says it uses their custom 'FireOS'. I was wondering if I could still use the kindle to debug my app because if not I will not purchase it.
Why it isn't a good idea
The Kindle Fire runs Amazon's customized version of Android, namely, FireOS.
FireOS has a very tight integration with Amazon's content distribution platform (Amazon's ecosystem, so to say) at the cost of the traditional Google integration we normally find in Android devices.
This is to say, Amazon devices, although they run Android, lack any Google Apps as default or installable from the Amazon AppStore. It is possible to root a Kindle Fire tablet and flash a GApps package, but that (IIRC) is a probable violation of Amazon's Terms of Service, so I will not include that information here. It can be found with a simple search, however.
How is this relevant to an app developer looking at a Kindle to test their apps?
If your app only relies on APIs provided by "core" Android, AOSP, there's no problem. Your app can successfully be used on Kindle Fire, and hence tested on it.
If your app relies on anything Google provides as OEM stock on most Android devices, such as Maps, Play Services or Play Games, your app won't run at all due to missing library dependencies, unless you have hacked the Kindle Fire and installed the core GApps onto it after rooting, flashing a custom recovery or ROM, etc.
If your app relies on Amazon Services, well, you should be using the Kindle Fire for testing. It is Amazon's product and will work best with it.
Finally, most Android developers do consider a Nexus as "the" device for Android app development and testing, but opinions differ, and you should decide your testing decice requirements on a case-by-case basis.
If your app is for android, you can use the android emulator what google offers to developers. The Amazon Kindle how Tamoghna says, runs a limited and custom system of Android.
If you cant run the google emulator you can try the Genymotion what offers to the users free virtual machines with android. Also offers a premium service to custom and more android versions.
The free version let you test and develop on the more common devices.