Linux is an open-source operating system that can be installed on a wide variety of hardware.  Most distributions (the term used for varieties) are free and very well documented.  There are two distributions that I highly recommend, but before we get to those I want to address the two questions above.

What is it?

As featured above, linux is an operating system.  That means that it is the part of the computer like Windows XP, Vista, 7 or OS X.  The majority of what is it will be answered below in the…

Why do I care?

You should care for a couple of reasons.  If for nothing else Linux is free, as in a free cup of coffee, and free as in the 1st amendment.  This creates a low cost highly secure and stable platform.  Linux is a very valid option for keeping an old computer running.  It does not come with any bloatware (junk that comes pre-installed on modern computers), which allows it to run faster with lower requirements.

Linux is also more secure for a variety of reasons.  The primary reason is that there is no push to write viruses and spyware for a platform used by only 1.3% of computers.

I use Linux based machines for data recovery because they are the only stable option that can read and write to ALL of the disks from ALL operating systems used in recent years.

Recommended Distributions:

I highly recommend ubuntu for most beginners because of the huge user base and the amount of support that can be found.  My personal favorite is Linux Mint.  Mint is built on ubuntu and includes basic add-ons that 90% of users will install anyway (flash plug-ins, etc).