Twitter recently changed notifications, allowing you to get emails about replies or retweets or other activity in your timeline. Since Twitter's default "Tailored for you" is mysterious at best and useless at worst, here's how to go adjust those notifications:
First, go to Twitter and log in. When you're logged in, you'll see your timeline in the middle of the page. Click on the gear in the top right corner, and then click Settings from the menu that drops down.
Why yes, that IS Neil Patrick Harris barefoot on a unicorn. Why do you ask?
Once you choose Settings, then click on Email Notifications on the left. From there, you will see the notification options. By default, these are all on and all set to "Tailored for you." Your only other option is "By anyone" so your options are either "everything" or "just what Twitter decides is best," so it's possible you don't have any interest in these notifications.
Here's Web notifications. And also NPH.
If you actually use the website for Twitter instead of a client, you can get Web notifications that pop up in your browser. In Twitter settings, click on Web Notifications and you can adjust the notifications you see in your browser.