Distro Orgy

How many are there?
As many as can be.

"Distro" refers to a Distribution, or a software package/bundle. Linux per se forms a small part of a Distro - the Kernel. The central part of an Operating system that manages the allocation of system resources to different routines.
It is the routines that are of interest to the user, since the Kernel is in most part not directly interacting with the user.
Routines take on different forms, most notably like applications like word processors, browsers etc.

Thanks to the open, free, cooperative community nature of the Linux movement, anyone can bundle along the Linux kernel and a handful of routines to create a distro.

It is an orgy when it comes to distro selection.

Distros can be classified into different types based on different criteria.

1. Liveness

A Live distro can be used without installation to the hard disk. Knoppix is probably the first such distro. It has many advantages, most important is the ability to test the distro without worrying about the effecting the existing data on the hard drive.
A Non live distro need to be installed before it can be used.
Many distros which were non live to start with are not available as Live versions as well. And a Live distro need not support installation to the hard disk, meaning you may have to download a non live version to install to the hard disk if you liked the live version. But this is a rare event.

2. Size
Distros can be slimmer and contain fewer routines making it possible to create a installation CD of sub 700 MB (Damn Small Linux is a distro that lives up to it's name by offering amazing functionality in a 50 MB installation size). Or distros can go the whole nine yards and pack in enough routines to fit a 4.7 GB DVD. Both cases are Live. So depending on the willingness to download stuff, or availability of a DVD writer, one can choose which distro image to opt for. If a CD is used, and the user needs additional routines, they can be downloaded on a case by case basis. This need not be that straight forward in practice, though.

3. Desktop Environment
The "face" of Linux is the user interface - the windows, the menus, the buttons etc. There are mainly two user interfaces - KDE and Gnome. Others also exists like IceWM. Most live CD distros are based on KDE or Gnome. DVD distros can pack in all environments. Gnome is a product that syncs well with the ideology of the free software movement, that also includes the Linux kernel. KDE on the other hard includes some non-free code making it principally incompatible with the free software movement. But this is just for academic purposes. Functionally, both UIs are good.

Some more popular distros are mentioned next.
1. Ubuntu, Kubuntu
2. Fedora Core, Red Hat Enterprise Linux - FC is free, RHEL is not.
3. OpenSuse - probably the free version of Suse
4. Linspire, Freespire - Linspire is not free.
5. Mandriva -
6. Centos - based on Ubuntu, live with difficult hard drive installation options.
7. Debian, Slackware - one of the earliest distros.
8 . Knoppix - Live with easy hard drive installation options.
There are many more distros, and many of those - including many on this list - are based on more popular distros like Fedora Core or Debian. "Based on" means repackaged : take a Fedora Core distro - modify routines - change default colour scheme - add new label.