The question is elementary, as we all know that all those feared black hat hackers would go to town on the opportunity of being able to access a PC remotely, but could someone really do something with an IP address?

Most people have a router with a firewall enabled, which deters most potential attacks, but even then, the IP address you may receive most likely points to the ISP that the person uses, since the ISP location itself assigns an IP address to the household.

So am I correct in assuming that in order to attain a particular machine's IP address, you would have to get the IP address of the ISP, then get the IP address that the ISP assigns that person, then get the home IP address if the person has a router?

And even if you could accomplish that, what could you do with that IP address? Could you connect to someone's PC as a remote computer on the network if it was unsecured, viewing files and the like? Obviously, you could sneak in malicious programs and the like...

Also, how does a Trojan make things easier in creating that "backdoor?" Does it automatically establish a direct connection to the attacking computer should the attacking computer exploit it?

Well, one remarkable thing IP addresses can be used for is location tracking. It's hardly an exact science, but is often remarkably accurate when it comes to general location anyway. There's an official IP address directory worldwide that can be found here, and uses such IP address tracking to present info about a computer based on its IP address including location:

http://www.arin.net/whois/

For me it shows the location a few cities off, but definitely in the general vicinity. From what I've seen the tracking is remarkably accurate, often in the same zip code. IP addresses also provide info about computer type and service provider, among various other kinds of info.

And some IP address trackers not only present the service provider, but also contact info so the provider can be contacted about potential hackers or spammers.

If somebody is using offensive security software designed to penetrate and test the security of systems, then I'm pretty sure they can do whatever they please.

as far as i know....they check for open ports on a specific IP....which can be easily found out
next they just need to know which service is running at that port....after that its their job ....well i m unfamiliar with this part ...
they use security loopholes to enter ur system....but its not that easy ...i wonder how guys do it.

Surely you can do many things, if this ip address is for a server, i (as a programmer) can create socket and connect to a specific service on this server provided that i know the port of this service....;every service on server has a specific port number, familiar services have certain known number; for example web service has 80...

well I heard from a friend that if u have the persons IP address then using some script that you would insert into a Command Prompt...(start>run>type cmd)

then u could eventually get access into that computers shared folders and from there u could do anything that you want.

Now im not sure if this is true but i looked it up breifly and i found out that u could access the shared folders but it said that u could only access the files in those specific folders.

Ips can be easily hacked by the hackers and so you may need to protect your ip address from such guyz ...so one thing you can do is to hide or change your IP address
You can just google for more info because googling gives you a lot of search results and posting all of them here isn't possible...even if i post here the moderators gonna kill me as they could consider me as a spam...