It's not just in Linux distributions. AMD64 is the "friendlier" term for the 64-bit x86 architecture (x86_64 or "64-bit Intel" as a lot of people like to call it.)
What this means in the context of Linux is that distribution is available for 64-bit PCs. This is akin to Microsoft labeling certain versions of Windows as being "64-bit." It should also be noted Linux came to x86_64 long before Windows did and arguably uses long mode in so many better ways than Windows (Including the fact Linux has far more of a selection of 64-bit software than Windows.).
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, and the purpose the distro is meant to serve.
This is a holding question for alternates dealing with long-obsolete Linux distributions, such as Red Hat Linux 9. Please do not split the alternates out. There is no reason to use these distributions; they no longer receive any security updates, may not run on modern hardware, and many modern Linux distributions are free.
There is no such thing as "Linux XP". If you're referring to Windows XP, then the answer is no. As far as Linux distributions, for the majority of distributions, they are free (as in freedom) and free-of-charge.
Most Linux distributions are free
Many Linux distributions are intended for home users.
"Distros", or distributions.
Distributions
Ubuntu Linux is officially supported on the x86 and AMD64-based processors, and unofficial support is available for Intel IA-64, and PowerPC.
Nothing. Most Linux distributions are free of charge. There are some distributions that are commercially sold, though those sales are more likely subscriptions for support, not for the Linux distribution itself.
the kernal is the central part of the Linux operating system and determines how the system works - all distributions of Linux are based on this.
The Linux kernel. Beyond that, there are numerous alternative implementations of virtually every Linux program.
Linux is open source, not shareware or proprietary. There are commercial Linux distributions.