What are the distributions of Linux and how are they categorized?

There are many ways to categorize Linux distributions. You can categorize them by their size, whether they run on a LiveCD, whether they are provided gratis, their ancestry of other Linux distros,...

A Linux distribution, known as distro or flavor, is an operating system that uses the Linux Kernel. I think the most common one is Ubuntu.

A Linux Distribution (distro) is lika a flavour. For example: You can go to a dairy bar and ask for ice cream, but you need to specify which kind of ice cream. Chocolate ice cream, strawberry, bubble...

Most modern Linux distributions have a method of updating the system without requiring a clean install. The exact procedure for this varies by distro.

Yes. BackTrack is a Linux distribution that focuses on penetration testing and computer forensics.