Error connecting to your database is common, but it isn't necessarily a problem with wordpress itself.
-First, double check your wp-config.php file settings for the database name, database username and database password. This is where most errors occur.
-Then check that you don't need to change the database host from "localhost" to something else. You can get that info from your hosting provider.
-Check you have actually created a database with the same database name as is in your wp-config.php file.
-Last, but not least, if all that information is correct your database probably has a problem with it, and you may need to contact your hosting provider.
There are also many threads on this topic on these forums: http://wordpress.org/search/Error+establishing+a+database+connection?forums=1