Switching Hosts, problems with the DNS server Options

[venti-anemoi]

I'm don't know how else to explain the problem other than recounting everything that happened, so prepare for a long post.

I won a domain in June through a contest, and the person who hosted the contest currently manages my domain. In the process of getting it up and running, my manager (for lack of a better word) asked me to get the DNS servers from my host and give them to her, which I did. She also said that she was the only one who could change the DNS server.

My site went down a month or so ago, and I contacted my host asking him if something was wrong. He told me that he had switched to a newer, much more expensive server, and that he couldn't host me anymore unless I was able to pay.

I found a new host a while ago who tried to fully host my domain, sylfaerien.net, but said that when she tried to host it, it said the name was "already in the server," meaning my old host needed to unhost my domain. She's using dreamhost, which I'm assuming my old host used before he switched.

So I contacted my former host telling him what my new host told me, and he said, rather rudely, that my host needed to change the DNS name, or if she had her own server, to change the A name. But my manager is the one who has control over the DNS server, so do I contact her first and tell her to change it? And if so, change it to what? I'm loathe to contact my former host again because he hasn't been the nicest of hosts.

In short, I'm utterly confused as to where the root of the problem is. If someone could explain the whole DNS server/name thing, and how it's connected to the domain (in short, how is a domain set up?) or direct me to a site that explains it, I would be eternally grateful. I tried googling it, but I have no idea what to look for.

Thanks for reading. :]