What is root?

root is the superuser in Linux operating systems and since Android is also a Linux based operating system, it is the superuser in Android.  Superuser means that it has complete control over the system files, memory, etc.  root user allows you to make changes that are otherwise denied using the normal (regular) user.  This is the way you should use your computer (well, it applies to all devices as it is a good security practice) and only use root when you need to make changes to your system to prevent accidental damages to your system.  root in the context of Kindle Fire also means that you are stuck with Amazon's Apps and way of accomplishing tasks.  some people are satisfied with the default stock and that's okay, not everyone feels adventurous to try rooting & custom roms.

What is stock Kindle Fire?

Stock is the default operating system for Kindle Fire without being rooted & without any modifications to the default Apps & so on.

Stock also means that you cannot use Google Play.

Kindle Fire stuck on boot screen (animation, logo)

Sometimes, the Kindle Fire gets stuck on the boot screen animation logo, there are various reasons why it would freeze like that, if your Kindle Fire is still vanilla (e.g. you haven't rooted it), it probably just needs a hard reset.

How to hard reset a Kindle Fire?

• Press and hold the power button for around 20 seconds and release.
• When it turns off, turn your Kindle Fire on by pressing the power button.
• That's it, your KF is running again! 

If that doesn't work, please check this article, it lists several methods.