There are two avenues here, I'll call them the "practical argument" and the "philosophical argument":
A major exponent of the philosophical argument is Richard M. Stallman. I can't hope to sum up all the many things that RMS has said on this topic[1], but some important ones are this:
We should have 4 software freedoms[2]:
0.) The freedom to run the program, for any purpose.
1.) The freedom to study how the program works, and adapt it to your needs. Access to the source code is a precondition for this.
2.) The freedom to redistribute copies so you can help your neighbor.
3.) The freedom to improve the program, and release your improvements (and modified versions in general) to the public, so that the whole community benefits. Access to the source code is a precondition for this.
So it is an espousal of community-building and sharing that this model tries to effect. Why does Linux exist? Because people built it and agreed to share their work with each other. Now that they've done that, we're all richer for it. It is estimated that[3] the cost to re-develop Linux from scratch would be anywhere from $170m to $1 billion.
And there is the argument from practicality. Linux lowers the cost of hardware. Not many people seem to appreciate the fact that when you buy a new computer, part of that price doesn't go into chips and hard drives, but to the OS vendor[4] (whomever this may be).
If you are a coder, you get an array of very good free compilers for many languages. GCC supports C, C++, Objective-C, Fortran, Java, and Ada. It is a safe bet that you can write in nearly any programming language under Linux (and yes, there is even a .NET implementation[5]).
So, if you're convinced try out some of the popular distros from distrowatch[6]:
* Ubuntu
* OpenSuse
* Fedora
* Debian
* Mandriva
* Mint
* PCLinuxOS
* Slackware
* Gentoo
* CentOS
I think Ubuntu or Fedora would be good places to start. They are each a little different from each other. One thing that comes to mind is that Ubuntu is a Debian-based linux and uses DEB packages whereas Fedora is Red Hat-based and uses RPM. These are systems for easily installing binaries, I prefer DEB myself, but they are pretty similar.
EDIT: Some random things that set Linux apart from Windows:
* You can run it as a "Live CD", that is you can pop in a linux cd and reboot your computer. You'll then be running linux without the need to install anything on your hard drive. In fact, there are distros that never use the hard drive.
* Linux is very flexible. There are versions that fit on a single floppy[7][8], a USB stick[9] all the way up to a DVD[10].
* Linux is secure, or at least much more secure than Windows. There are ways to partition your disks, use permissions wisely, and chroot jail programs. This all amounts to providing a very robust and secure system (OpenBSD also addresses some of these issues, but you asked about Linux :) )