Android/iOS: Recently, Twitter has been rolling out some extra notifications on its mobile clients that you may not really care about, like "People are talking about this" or "Three people you follow just followed this guy". If you'd like to get rid of these, there's a setting for it, but it's buried.
On both iOS and Android, the fine-grained notification settings are tucked away under less-than-obvious menus. As Mashable points out, on iOS, you first open Twitter's main settings panel and tap on the account you want to change. Scroll down far enough and you'll see options for notifications for things like "Recommendations", "News" and "Other." While none of these are very obvious labels, they're also not direct interactions with you, which we assume are what you're most interested in.
Android's layout is even more confusing. Under Settings, choose the account you want to modify. The top option is "Notifications" and underneath it says "Notifications are ON." One would assume this is a toggle, but tap this button and you'll find a list of all the notifications you can enable/disable. Scroll to the bottom to find "Recommendations", "News", "Other" and any of the other types of notifications you might want to disable.