When someone goes to your website (say http://www.yournewsite.com), the domain name gets resolved to an IP address. This IP Address is that of the web server where you are hosting your website.
This automatic resolution of domain name to IP Address happens by DNS lookups and the original record mapping of your domain name to the public IP address is maintained in the Name Server.
So after you point your domain's NS records to ns1.itahost.com & ns2.itahost.com, the requests to your domain name would reach ns1.itahost.com or ns2.itahost.com where it would be resolved to the public IP address of the server where your application is hosted.
Eventually, the mapping of domain name to IP address would get copied on to different DNS servers across the world, but during a lookup if any DNS server cannot find it, it could trace it from ns1.itahost.com or ns2.itahost.com because that's the NS record for your domain name.
A bit of extra information:
In the name servers (ie ns1.itahost.com ns2.itahost.com), you keep different record mappings for websites, sub domains, email etc. I guess this activity will be done by the hosting provider, but you could check with your provider.
If need be, then you could host your website under one hosting provider and email under another. In this case, you'll have to map the A-record (for website) and MX record (for email) to the respective IP addresses.
Hope this helps. Cheers.