OS X Security - What is a local subnet?

In System Preferences>Sharing, below "Computer Name:" it says "Other computers on your local subnet can access your computer at (name withheld)" What is a local subnet? Should I be wary of this? How do I shut this off? The Edit button only allows a change of name.

A local subnet would be a network where all network-attached devices (computers, printers, etc.) are behind a firewall/router and have a private IP address in the same range (i.e., 10.0.n.n, 172.n.n.n, or 192.168.n.n). For an example, in a home network with two computers and a printer attached to a switch/hub, each of those attached devices would be on the same local subnet. Because they are all on the same local subnet, it would be easy for them to communicate with each other for things like sharing files. A local subnet is one of the primary features of networking...nothing to be wary about. You can "shut it off" by not connecting your computer to a network...or turning off the Ethernet/AirPort ports.

Also, does the Enable Stealth Mode work without the Firewall turned on?

No.

"What is a local subnet?"

http://www.webopedia.com/TERM/s/subnet.html

"Should I be wary of this?"

No.

"How do I shut this off? "

You don't. Every networked computer in the world is part of a subnet.

So then a subnet has nothing to do with computers that aren't in my house as a part of my personal computer set-up?

Correct.

There's only one computer hooked up behind the firewall -- the computer that I'm using -- a printer and a scanner. So no one outside of that can access my stuff with their computer through the subnet?

As long as you have your firewall enabled, specifically the firewall on the Internet router, those outside of your local network will not be able to access computers/printers within it.