There are a few ways (depending on class structure). However, your class is probably going to be using syntax similar to:
$DB = new YourDatabaseClass(...);
$DB->query("your query");
and so on from within your other classes. If you only need / have the one database or database object then you can go about passing in that database via constructor (a function within your class with same name as your class that is called when you class is initialized).
I am really not sure what else you are going after, so here is a basic example of what I am talking about (although you can approach this in many different ways). The one thing I would discourage is the use of globals (such as using "global $DB;" ). Also, the style below should be PHP 4 friendly.
If you like I can also post a database class that I use, just ask.