It's pretty straightforward and I advise you untick all the options, definitely if you have a twitter client with push notifications.
- Go to Twitter.com
- Click on Settings
- Click on Notifications
I don't understand twitter, because right now millions of users are getting a ridiculous amount of emails that they will not want. I would love to find out how many people will turn this option off within 24 hours, the only thing that comes to mind is that new users who don't get many mentions, rarely gets tweets favorited and when someone retweets them.
I like that they add features that people might use but turning them on by default is a bit lame. Oh well you can see above mine are all switched off so won't be visiting twitter site again for a while.