apps > twitter

The Twitter mobile app has started sending push notifications for things that you didn't sign up for. Some people will view these as spam, especially because the settings to disable it are buried.

Go to Settings > your account > Notifications

When you first see the "Notifications" item, you might not click on it because you don't want to disable all notifications. Actually, it's not really a checkbox, it's a submenu:

Favorites Tweets from starred accounts. Enable this per-user by tapping the yellow star on their profile page.
New followers
Messages Direct messages.
Twitter Alerts Emergency alerts only. It's a good idea to keep this.
New contacts Recommended accounts?
Recommendations Recommended tweets. Annoying.
News Breaking news.
Other Other Twitter experiments, like when your friends talk about TV shows.