Zen Cart's connection process to the database is very simple:

1. You supply the database name, username, password, and host
2. Zen Cart uses that information to make a connection to the database using the mysql_connect() and mysql_select_db() functions built-in to PHP.

If the MySQL engine sends back an error response when trying to connect or to select access to that database, then Zen Cart will give the error message you are reporting.

So, that means any of the following, or all, could be true:
a) you supplied wrong information for database name, username, password, or host.
b) you haven't granted that username all permissions to the specified database

See the /docs/ImplementationGuide.pdf file for more help on what is required when setting up a database for use, or at least guidance on what help you will need from your hosting company if you're not comfortable with the information presented therein.