Linux is a buzzword which has been circulating on the edge of the mass consciousness for several years - the name is kind of familiar but very few PC owners have a clear idea of what it is. A common idea is that it's a "geeky" alternative to Microsoft Windows, used by a minority of rabid Microsoft-haters for no other reason than because they can. It's perceived to be altogether far trickier, more complicated, and just more hassle than the operating system that their PC came with. It's just for techies who enjoy fiddling with the computer to persuade it to work, not for the average home user.
Some people may have heard of unix, and associate all such things with unfriendly, arcane command windows, where you have to remember complex commands and type them in, rather than use a 'normal' graphical interface, with icons to click on and menu bars to show you what you can do. Fortunately, yes, Linux does (normally) come with a desktop, or rather a selection of various desktops, many of which look surprisingly familiar with a task bar, application windows, toolbars and menus, mouse pointers and all the things you might expect.
Without diving too deep into the technical terminology about which bits are Linux and which bits are built on top, the basic point is that with a Linux-based system, you can get a fully-functioning, incredibly powerful, intricately-configurable and very robust system which allows you to do a spectacular number of things. Many people have both a Microsoft system and a Linux-based system installed on the same machine - at boot time you can choose which system to use. But many people also find that they don't use the Microsft system any more, because they can do everything -- and much much more -- with the Linux system.
There is much discussion about 'free' software and what it means to be free. Philosophy aside, there are two standard interpretations of the word 'free' - one relating to beer and one relating to speech. Both meanings can also apply to software, and to Linux.
"Free as in beer" means that you don't have to pay any money to receive it. It is possible to just download a version of Linux and use it without paying anybody anything, and this is perfectly legal and by design. However, you can also pay somebody money and they will provide you with a nicely printed, glossy version, and offer support for you and maybe other services. So Linux is not always cost-free, and there are many companies offering products and services based on Linux.
"Free as in speech" means that the licences associated with the software are such that the freedom of the software source is protected. Unlike some other software, where the source code is carefully guarded by a single company and only the compiled, runnable code is allowed out to the outside world (where copying and distributing it is restricted), the source code of this kind of software is available for anyone to see. Anyone can copy the software, even modify it if they like, and redistribute it, as long as they pass those rights along to the next person. The whole open source software movement is based on the premise that if you make the source code freely available, a community develops around it to improve it and feed the improvements back.
Therefore you can have software which costs no money but has no freedom (for example, Winzip or Winamp), and you can have software which has freedom but costs money (for example, SuSe Linux). Some open source software runs on Windows (like Firefox), and some closed-source software runs on Linux (like Adobe Photoshop).
There are many reasons people give for switching to a Linux system, but among the most popular reasons are increased reliability and security of the system, especially resistance to viruses, worms, trojans, spyware and the like. One can debate whether this is inherent in the systems' designs, or quality of implementations, or whether it's just because the number of Windows machines presents a critical mass of targets to the malware authors and the number of Linux systems is (currently) so low as to make them not worth the programming effort to attack. Point is, such malware is a real problem on Windows systems, and much less so on Linux systems.
Another important benefit for some people is the trustworthiness of their system. A lot of sensitive and important data flows through the system, so it's important to have trust that the computer is indeed benign, that the system is oriented around the user, and not just around corporate profits. Let's say, hypothetically speaking, a computer system were to keep a log of which commands were run by a user and which keywords they were searching for among their files or on the internet. And let's say, hypothetically speaking, you noticed this and wanted to know what it was doing. If this were an open source operating system, you could find people to ask, search the newsgroups, post questions, and read into the subject to whatever depth you wanted. You could even read the source code, and see exactly what was happening. You could most definitely switch it off if you were concerned about it, and more than likely configure its behaviour to how you desired. In short, you're in control of your computer.
Now let's take the same example, but assume that it's not a Linux system behaving in this way, but a closed-source system, which you have bought. Now what can you do? You can of course search the internet and post questions to see if anyone else has managed to find out the answer, but the company that sells the system doesn't have to tell you what it's doing and it's clearly not in their interests to do so. It's not in their interests to let you turn it off, either, and they're certainly not going to allow you to change the software yourself so that it does what you want it to.
It's also true that many people are attracted to experiment with Linux just to see what it's about - how can it possibly work if it's free? How can so many people just donate their free time to making something like that? Can it really be a serious alternative? The community that grows around it is also attractive - questions and tips are exchanged by email and newsgroups with such a friendly air of camaraderie and mutual support, one quickly gets the feeling that this is a powerful idea which can only grow and gather momentum.
Businesses repeatedly perform studies to examine the cost benefits of switching to Linux, balancing the initial purchase costs against support costs, retraining costs and the productivity drop while staff become accustomed to the new tools. For home users the situation is much more clear-cut, because Microsoft's "deals" with the PC manufacturers ensure that it's very difficult not to buy a Microsoft operating system - you're virtually forced to buy one even if you want to run Linux. Therefore even though the Microsoft system costs money and several Linux systems are completely free, the cost probably isn't one of the major incentives to switch.