Zone Alarm is a GREAT firewall.
If I am right the free version also tells you what ip, and or address is trying to access or ping your pc.
If your cable modem has no power then no one can access your pc. If your cable modem has power but your pc is turned off meaning it has no power then they still can not get in.
If the PC is on, and the modem is on yes they can access your pc.
Broadband connections are different from Dialup in more then just speed.
Broadband you have an ip "leased" to you in some cases. meaning the number that id's you (example 12.3.4.567) can be the same number for a few weeks to months.
Dialup your ip changes everytime you "login or dialup" your isp
So 12.3.4.567 may be your ip now. Disconnect for some reason then reconnect an hour later your ip could be 23.5.6.912
Which that makes it harder for hackers who are trying to get in. On broadband the IP doesnt chage that often so if you turn your pc off, and think in an hour you can go back on without the hacker trying to get at you. You will be surprise he/she will still be there trying your IP
A router with a built in firewall is nice to have (I run one at Geek HQ) I also have zonealarm running on all my pc's connected to the router. Why that much security you ask. IM PARANOID
Basically just practice safe net usuage. If you have a broadband connection by all means run a firewall. Make sure you have an anti-virus software installed on your pc also, and it is up to date.
If you want to share your broadband with multiple pc's in your home go with a router ( I have a linksys ) just remember if you have any problems your ISP will only trouble shoot up to the modem they will not trouble shoot the router etc since technically you are breaking one of their rules. WHO CARES
I know theres allot to read there but I figured I would add my 2 cents for what they are worth