Umm. It took me a while to get my head around it, so I might not be the best at explaining it lol!
Basically, it's what your router connects to when you type in an website, and it matches it up with the ip address for that particular site. So then it can take you there. ^^
If you want to get rid of that Verizon search site when you type a word, you'd have to access the router settings and change which DNS servers you connect to, for instance to the OpenDNS one I mentioned.
It sounds pretty complex but in reality it's just changing a few numbers, and the end functionality remains mostly the same (apart from some quirks as you've noticed).