DHCP is almost universally used to obtain a Local IP Address. Private Routers assign a Local IP Address usually in the 192.168. 1.xxx range.
Some ISPs provide Modems that are designed to attach one computer, attaching a router to the modem allows more computers to be connected.
Some ISPs provide Broadband Routers and multiple devices can be connected.
SOME IPSs provide Modem / Routers that do NOT allow hosting websites or databases on the attached computers.
No matter which you have, the Device from the ISP gets a Global IP Address. You can view yours at http://whatismyipaddress.com
This Global IP Address can be paid for and unchanging (Static) OR free and possibly change at random (Dynamic) (often during electrical outages)
The router hands out Local IP Addresses to each attached Device - computer, pad, or phone.
If you have multiple computers attached, you can view each Local IP Address in the network settings control panel (Mac or Win or Linux)
The IWP request from someone on the internets can come in on one of many ports.
Port 80 and Port 591 are the most commonly used Port Numbers for Filemaker Hosting.
The internet uses the Global IP Address to talk to the Modem / Router. The router uses the Local IP Address to talk to it's computers.
When someone sends a request for your hosted database, the router has to know which computer to send the request to AND which computer to relay the answer from to the internet requestor. This port forwarding on the router is setup by going into the router's setup. http://portforward.com explains this.
This Google "time warner forums" https://www.google.com/search?q=time+warner+forums Lists the Forums to ask questions about their service.
This Google " iwp site:forums.filemaker.com" https://www.google.com/search?q=+iwp+site%3Aforums.filemaker.com
Lists the threads about IWP on these forums.