Well......
Port Scanning is one of the most popular reconnaissance techniques attackers use to discover services they can break into. All machines connected to a Local Area Network (LAN) or Internet run many services that listen at well-known and not so well known ports. By port scanning the attacker finds which ports are available (i.e., what service might be listing to a port). Essentially, a port scan consists of sending a message to each port, one at a time. The kind of response received indicates whether the port is used and can therefore be probed further for weakness. Basically, it's like ringing the doorbell to see whether someone's at home. A port scanning program that you install and run will tell you what ports are "open" and if they present a viable target.
Not sure your info about cable modems being a good target is too accurate. Cable service generates dynamic IPs, not static, and the modem is a rudimentary firewall in and of itself. I also installed a router/hub that acts in a similar manner, as well as interior firewall setups and scanners, but I'm a paranoid bastard and love to hunt down anyone pinging my system and give them grief. There's tracking programs for such things as well, although you're getting into more advanced areas at that point, and eventually you'll be walking a fine line in terms of "returning fire"....I've been accused a few times of being the source of denial of service attacks, mostly because I have the hardware to launch them. Programs that go beyond the norm tend to be expensive and difficult to install or maintain. If you're on a dial up service and turn your computer off when not using it, I'd say you have little to be concerned about other than the viruses running around.
One other word of caution...most computers are "taken over" when someone gets physical access to them. Be careful who you allow to putter around on your rig....it only takes a few minutes to install the necessary software (no details...we're not trying to breed hackers here, ya know) and be able to remote someone elses computer.
Email me if you have specific areas that you don't want discussed on the open forum. If I don't have the answers, I have low friends in high places that would... Q