In order to make a more compelling argument I've separated this guide into two main sections: reasons to use Linux on a server and reasons to use Linux on a desktop. Some over lap so I encourage you to read both sections.
Linux As A Servers
No expensive licensing fees
Unlike Windows and Mac OS X, every Linux distribution is provided free and can be used or extended as you or your business see fit. Typically this frees from $100 - $4000 dollars for you to spend on something else.
Supports extensive number of server protocols: http, smtp, imap, pop3, samba, nfs, afp, etc...
Linux, through the help of the Open Source Community, offers almost any server you may need to run, and in some cases the Linux server is more reliable, more efficient, and more popular than it's Windows counter part. (Please note that Mac OSX also uses many of the same servers that are available to Linux users.) Here's a list of some of the server protocols supported by Linux: http, smtp, imap, pop3, samba, afp, ftp, ssh, telnet, sip, mgcp, and many others.
Added On 2006-03-07:
As requested here are some links to great server products
Secure out of the box
Linux is naturally secure out of the box. Most distributions come with a suite of security features, including a firewall, IPSec, and Security-Enhanced Linux, which is a super secure version of the Linux kernel developed by the NSA. In addition to its security features, Linux is also much more resilient to attach by viruses and worms, and tends to receive security fixes much faster than other operating systems.
Enterprise support contracts available
There are a large number of companies that offer support for Linux; including IBM, Novell, and RedHat. The large number of choices can enable your business to find a solution customized to its exact needs.
Large friendly communities
In case you can't fork over the money for professional support, you can turn to the open source communities. These are usually bulletin boards, email lists, or IRC chat rooms where groups of users congregate to help each other with problems and make the software they all use better. Everyone is encouraged to participate, and it's free.
Many individual variations available (distributions)
There are hundreds or even thousands of variations of Linux available. They come in all shapes and sizes, and they all do things differently. This is great for the user because now you can find the one that fits your needs the best. If you want a full featured large distribution, try Suse, Fedora, Mandriva, or Ubuntu. Want a CD distribution, a USB memory stick distribution, or even a distribution that runs on an Linksys Router? Because distribution for all of those currently exist. For a large list of current Linux distributions check out DistroWatch.
Configurable to your exact specifications
Let's just say that after checking through hundreds of distributions you still can't find one exactly to your specifications. Then you are entitled to change an existing distribution or create a new one to fit your exact needs.
With the increase in cheap, powerful hardware server virtualization is becoming an attractive way to maximize the use of equipment. There are a few open source project devoted to making this available on Linux.
RAID & High Availability
Linux has support for a large variety of RAID hardware and kernel support for software RAID. In addition there are several projects working on high availability servers through server clustering.
Linux As A Desktop / Media Center
No licensing fees
The reasons are the same as listed above, who wants to pay for a copy of Windows XP which is FIVE years old!
There are open source equivalents to most commercial software
Most commercial Windows products have a open source equivalent. Instead of using MS Office, you can try Open Office. Instead of Dreamweaver, you can try NVU. Instead of Internet Explorer, you can use Firefox. Instead of Windows Media Player, try XINE, MPlayer, or VideoLan Player. Instead of iTunes, MusicMatch, or WinAMP; try Amorak or XMMS. There are also a host of instant messaging, email, and other programs. The list just goes on.
WINE can run a large amount of Windows software
In the limited cases where there is no Linux equivalent, you can always try using WINE. It is able to run many popular Windows applications natively in Linux. Check out their website for performance comparisons and a full list of applications known to work with WINE. WineHQ.
Software you don't get with Windows
In addition to doing what Windows does, Linux does what Windows doesn't. It provides you with the source code and all of the tools needed to change it. You get several great programming languages and development environments for them. If you're an aspiring computer student you can't pass this up.
Excellent multimedia support
Linux has excellent multimedia support. Thanks to the work of many daring open source projects you can watch or listen to almost all formats of video or audio. In addition to decoding these formats you can also encode many of these format, which is nice if you have any interest in video / audio editing.
Linux has many of the great features of other modern operating systems including support for 32 and 64 bit processors, virtual memory (swap), advanced memory management and file buffering, multi-threading, preemptive multitasking, symmetric multiprocessing, real time process scheduling, several advanced journaling file systems, and many other great features.
Secure out of the box
As I mentioned above, Linux is secure out of the box. It comes with native security software and is much more resilient to viruses and spyware / adware than other popular operating systems.
Ease of Use
Check out some of the latest distributions from Suse, Ubuntu, and Mandriva. They are very slick and easy to use, easily performing the tasks of normal computer users much safer and faster than other operating systems. Firefox for web browsing, Thunderbird for email, plug-in your digital camera and hotplug recognizes it with no configuration, type papers, make spreadsheets, and presentations with Open Office, and listen to music with Amorak or XMMS. I can't think of any more things an average computer user would do, but I'm sure other people have and they have already included software to do it with Linux.
Still not convinced that you should use Linux? Then you probably work for Microsoft or Apple. But seriously, try a Live CD distribution link Knoppix (all you have to do is burn a CD, and reboot your computer, it doesn't disrupt Windows / Mac OS at all) and see for yourself.
Please leave your comments below. Feedback is always appreciated.