I know you can turn off notifications from the native email client, but is there a way to turn them off for Gmail, Google+, and other Google apps? I have my Google account set up on my phone, which is useful, but if I can't specifically turn off Google notifications I'm going to have to unsync it, because as it stands I have to either silence all notifications and miss important text messages, or get woken up by G+ comments and NYT emails at 4 AM.
To turn off notifications for Google+, open the Google+ app, press Menu->Settings->[Your account name]->Notifications, then press the ON/OFF toggle in the top right. In recent versions, notification settings are per-account.
To turn off notifications for Gmail/Google Apps Mail, open the Gmail app, press Menu->Settings, select the account you wish to disable notifications for, uncheck the checkbox that says: Email notifications
To turn off notifications for Google Talk, open the Google Talk app, press Menu->Settings, select the account you wish to disable notifications for, press IM notifications and select off, and the same for Video chat notifications
To turn off notifications for Google Voice, open the Google Voice app, press Menu->Settings->Sync and notifications, and uncheck the boxes for text and voicemail notifications.
NOTE: This is based on Android 4.0+, some of these apps (especially Google Talk) may function differently on pre-ICS.
If you are on Jelly Bean and rooted you can use an app I built to manage notifications for all applications from a single place.
Notifications Off allows you to turn off notifications for all applications. It can also automatically disable notifications for new apps as soon as they are installed.
The next version will have support for profiles, tasker integration and a widget for switching profiles.