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Is this a wordpress.com site or a self-hosted WordPress install?

A wordpress site.
Sorry am new using wordpress.

A wordpress.com site or a self-hosted WordPress site?

A wordpress.com site

I am hosting it on hostgator.

If it is a wordpress.com, you cannot be hosting it on Hostgator – although your domain name might be registered there. Did you install WordPress yourself?

I set it up on hostgator. It is hosted on hostgator not a wordpress.com site.

So you installed WordPress yourself on your Hostgator hosting account?

Using hostgator quick install. Does it take time after installing it on the ost before it comes up?

It can do. When did you register the domain name?

domain name registered a long time ago but just forwarded the name from godaddy to hostgator and just created the wordpress account on hostgator.

When did you set up the forwarding. Name server changes can take up to 72 hours to propagate fully.

I did barely 2 hours ago

Then I’d suggest waiting a couple of days to allow your DN changes to propagate.