Changing the domain Name Server (DNS)

With your domain name registered and hosting account set up, the final job is to tie the two things together so that when people type your domain name or URL into their web browser, The Internet Company immediately 'point' that query at your site on your Hostgator account.
Incidentally, although you probably noticed when you are registering your domain name that The Internet Company also offer web hosting, the majority of online marketing experts recommend that you should never register your domain name and host your account with the same organization.
Whilst you can be pretty confident that doing so is not going to represent any kind of problem 99.99% of the time, you are nevertheless entrusting the whole of your business to one organization if you use the same company to register your domain name and host your account.
Consequently, if for whatever reason, the company concerned did have problems, your business could be wiped out by their difficulties. In this scenario, it is not impossible that your site would disappear and your hosting account might come inoperable for an indeterminate period of time, meaning that you are out of business with no idea of when you are likely to get back online.
This is obviously not a risk you want to take. It is therefore the reason why you should register your domain and set up a hosting account with different organizations, which is in turn the reason why you need to change the DNS.

In order to do so, open up your domain name registry account, find the specific domain that you want to make alterations to and click on it. When you do so, you should see a 'Nameservers' button at the top of the page that looks something like this:

On the next page, take the information that Hostgator will have sent you by e-mail concerning their own nameserver and copy/paste those nameserver details into the appropriate boxes on the detail change form as indicated here (you almost always need to change two nameserver addresses):

Click on the orange 'OK' button and you should see a message that tells you the changes will be made in the next few hours.
Generally speaking, it is never likely to take more than 24 to 48 hours before these changes are propagated although in my own experience, it generally tends to happen far more quickly than this.
Once these changes are made, you are now in a situation where anyone who types your domain name into their browser will be immediately redirected to your Hostgator web hosting account where they will see your site.
However, there is obviously nothing on that site at the moment, so creating your site is the next step.