Ernie1 had the best reply, you can use the constructor to assign some necessary properties and even get it to fire off some methods for you.
He also dealt with the thorny issue of "what happens if all does not go as planned?".
$this->databaseHost = $databaseHost;
One feature of classes is encapsulation - so you can just as easy use:
$this->host = $host;
Anyway, you've declared them private - so they will never be seen/used anywhere else.
Its in a class called mysql so I'd have thought that $database prefixes is just extra typing, extra chance of getting it wrong and less readability. But others may disagree.