For years, Twitter's two-factor sign-in process has required SMS at some level, even if just as a backup. That's all well and good on a phone, but what about when you're on a tablet, or are in a situation (say, traveling abroad) where you'd rather not get a text? You're set from now on. Twitter has added support for verifying your sign-in exclusively through a third-party app. You still need a phone number to get things started, but software like Google Authenticator and Duo Mobile can now fill in after that, with no SMS fallback. The setup process is relatively straightforward -- the biggest step is scanning a QR code to produce the verification number you need.
This isn't really necessary for most people. It may be helpful if you're regularly signing in and out, however -- say, if you're a social media manager or just tend to switch devices on a frequent basis. Apps like Duo Mobile may only save you a few seconds, but that can quickly add up. This might also encourage you to use two-factor authentication if you really, really dislike the possibility of having to use SMS, and it could prevent account breaches that take advantage of phone number hijacks.