Linux is an operating system
For most people, Linux is either something they could use instead of Windows or instead of Mac or it is something they're running on their phone and calling Android.
The first question that usually gets asked after that explanation is "What is an operating system?" The answer is (technically) that the operating system is the core software controlling how a computer communicates with hardware and manages programs. That doesn't make sense to most people, so the answer I like instead is "Imagine a cab is like a computer. The operating system is the driver. He figures out how to get you where you want to be, worries about gas and maintenance and traffic laws so all you have to worry about is the price and destination."
What is Linux like?
Linux is like Windows, or OS X, or AIX. If you want a computer where you select programs from a menu, run programs inside of boxes that make you think of little windows and do things like read web-pages, then that's what Linux is like.
If you want your computer to run a database that everyone in the office can connect to, then that's what Linux is like.
If you want a computer to run server software that builds web pages for a hundred thousand users a minute and distributes that work between fifty different machines, then that's what Linux is like.
If you want a really big network drive that everyone in the office can store things on and a server to share all the printers out, then that is what Linux is like.
Android is a widely used but kind of special case:
Users who have Android phones don't usually even realize they are using Linux. Since Linux is usually a system with a lot of other software thrown in for free with many, many options, Android phones don't seem like Linux to even regular users. Nonetheless, what makes Linux is a tiny little piece of software at the center of everything that decides how to talk to the hardware and how to handle the software and that's the part Android uses. The reason Android is special is that it uses many little programs that aren't free or don't follow the tradition of being open for modification like most types of Linux do.
There are many types of Linux to choose from
While you can only pick from a handful of types of Windows or OS X versions to purchase, you can pick from dozens of well known and hundreds of lesser known versions of Linux.
With Windows, you can purchase a computer with Windows 7 or Server 2008. There are a couple possible extra options, you might purchase an older computer with Windows XP or Vista or Server 2003 and you can choose between Windows Small Business Server, Professional, Home or Starter editions. They all basically look and work the same but what they can do and how much they cost determines which is the right choice for you.
With Linux, you have dozens of overlapping choices no matter which purpose you need filled and dozens of options within those choices. To keep it really simple, here are some of the most common use scenarios and popular choices
Regular desktop for homework, papers and browsing the Internet (sort of like Windows Home edition uses)
Small Office network server to provide printer connections, network settings and file storage (sort of like Windows Small Business server)
Large office network server, database or web server (much like Windows 2008 Enterprise)