Maybe I've just got notifications turned off for other apps, but on Gingerbread, I only get notifications for texts, emails and updates.
As for everything else, apps that want to send notifications are forced to ask for permission the first time you open them. Presumably, you'd want to grant this permission to Words with Friends, since otherwise, you never know when you have to play :-)
But other than that, you just hit "no" every time, and that's that.
Weathers and stocks are "widgets" (except only those 2 are available and they can only be displayed in the notification center).
Aside from that, I'm not sure how it works on Android but on iOS an application must explicitly register for notifications the first time it's opened and this launches a dialog asking the user what kind of notification it allows.
So it's quite easy to tell applications to buzz off and not notify anything.