If you'd like to receive Twitter notifications to your device, you can enable push notifications in your Twitter app settings, or attach a phone number to your account for updates via SMS. Below are instructions for enabling Tweet notifications.
If you already have a phone number attached to your account:
You can receive Twitter notifications in the form of text messages. More information about using Twitter via SMS can be found here.
If you don't have a phone number attached to your account:
You can add your number by visiting https://twitter.com/devices. After you attach a device, you will then be able to opt into receiving mobile notifications from users you follow. Use the steps listed above.
More detailed information about push notifications, including screenshots, can be found in these troubleshooting articles: