Linux is an operating system. An operating system is the basic set of programs and utilities that make your computer run. Some other common operating systems are Unix (and its variants BSD, AIX, Solaris, HPUX, and others); DOS; Microsoft Windows; Amiga; and Mac OS. If you're interested in learning about some not-so-common operating systems, take a look here and here.
Linux is Free Software. Now, just because it's Free, doesn't necessarily mean it's free. Think "free" as in "free speech," not "free beer," as we in the Free Software/Open Source community like to say. In a nutshell, software that is free as in speech, like Linux, is distributed along with its source code so that anyone who receives it is free to make changes and redistribute it. So, not only is it ok to make copies of Linux and give them to your friends, it's also fine to tweak a few lines of the source code while you're at it -- as long as you also freely provide your modified source code to everyone else. To learn more about free software and the major software license it is distributed under, called the General Public License (GPL), go here. In addition to the GPL, there are many other software licenses that allow you to modify the source code. The Open Source Initiative approves these licenses and keeps a current list of them.
Linux is not owned by anyone. One misconception many first-time Linux.com readers have is that this site, Linux.com, is similar to Microsoft.com, which is owned and controlled by the company that produces the Windows operating system.
Not so!
No one company or individual "owns" Linux, which was developed, and is still being improved, by thousands of corporate-supported and volunteer programmers all over the world. Not even Linus Torvalds, who started the Linux ball rolling in 1991, "owns" Linux.
(However, the trademark "Linux" is owned by Linus Torvalds, so if you call something "Linux" it had better be Linux, not something else.)