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.
Mandrake was a very old Linux distribution. It's name was changed to "Mandriva" back in 2005.