You should make sure you use a password to secure your router. Just follow the instructions provided by your Internet Service Provider or router manufacturer to set your own password for the router instead of using the router default password.

early-
Are you saying that you use a computer now that has personal stuff on it, but that it's not connected to the Internet?

When you set the security protocol make sure you use WPA2 not WEP. The latter is not very secure. We have a neighbor that uses an R-rated name for his SSID and I was curious who he/she might be. Took me less than 5 minutes to break into his/her network which was secured with WEP. I wasn't using software that was really designed to hack and I have no previous experience doing it.

"I was curious who he/she might be"

Did you give any thought to maybe knocking on the door instead of breaking into their network to answer your curiosity?

"curious who he/she might be"

He didn't know which door to knock on?

Now they hit you with probably theft of service and other cybercrimes depending on the courts prosecutors eagerness, for hacking into electronic devices... heh

mike_kaiser is correct. WPA2 is safe, assuming you don't choose a network password like "hello". (I suggest choosing 5 or 6 words that you can remember and make a sentence out of them).

If given the choice, use "AES" instead of "TKIP". (The details are pretty geeky).

I didn't know what door to knock on and to the best of my knowledge there's no way to triangulate on a WiFi signal using an ordinary computer. Right now there are 12 wireless networks around me, plus my own.

I wasn't looking for any personal information but rather something like the name of computers that would point to the owner. For example, if the neighbors are Joe and Sally and the offensively named network has two computers named Joe and Sally...

It does illustrate how insecure WEP is though.