I have a domain and the settings for it are pointed at my IP address. But when I look up my IP address, it does not show my domain as the hostname. What settings do I have to change to make my IP translate into my domain as the hostname?

You have to talk with whomever you get your IP from and either give them the information you want the reverse ip to be or delegate the reverse to your dns server so you can provide the reverse information

I contacted my ISP just now and they said that they don't support networking issues and I have to figure it out on my own. Am I supposed to change something in the router settings? I don't see anything I can change...

Reverse dns isn't something you can change on your own unless you are the delegated entity for the reverse information on your ips. You can look up on arin, ripe (europe), or apnic (asia) your ip and see who the delegate is for your ip. The whois command should be able to give you enough info to see where you need to look further. One of the pieces of information is a contact for that network block, I guess you could send them an email asking what you'd need to do to update your reverse ip information with them.

for example if your ip was 75.126.162.205

whois 75.126.162.205
shows you it is delegated to softlayer technologies and the contact you want is ipadmin@softlayer.com

If your site is hosted by some company you might not have a dedicated ip for your site (named based virtual hosts in www) in which case it'll probably never get changed