Database Error just after installation

Hi. I have just installed wordpress through fantastico in hostgator cpanel. When I go to login to the wp-admin area, it comes up with an error message.

Error establishing a database connection

This either means that the username and password information in your wp-config.php file is incorrect or we can’t contact the database server at localhost. This could mean your host’s database server is down.

* Are you sure you have the correct username and password?
* Are you sure that you have typed the correct hostname?
* Are you sure that the database server is running?

I don’t know how to change anything if I cannot get in, however, the username and password is correct, yet it doesn’t give me an opportunity to put them in.

I am wondering how long it takes once wordpress is installed to actually work??

The domain name is http://enlightenedeagle.com.au/blog/wp-admin

I am a newbie, so would love some help with this. I actually have installed wordpress on another domain and had no problems at all.

many thanks Barb