A Twitter App is an application that takes advantage of the Twitter API to leverage Twitter’s massive stream of data, often called the firehose, and deliver a customized experience for the app’s user.
Apps made by Twitter for use on mobile or tablet devices are called official apps. There are hundreds of third-party vendor apps as well.
For instance, TweetDeck and HootSuite are major apps that sit on top of not just Twitter, but Facebook, and other social media platforms that provide all the feeds on one screen for easier manageability.
The big idea of using some kind of Twitter app is to make Twitter more interesting to use.
For example, TweetDeck and Hootsuite (amongst others in their niche) allow you to manage Tweets in more dynamic ways than you can on the web. For example, they allow you to schedule Tweets and make reports.
Other apps do automated things for you, like letting you know who unfollowed you recently, or sending you a keyword report every day letting you know who mentioned you. Others will help you cheat when trying to get new followers, send automatic direct messages to new followers, or will get you to unfollow anyone who unfollows you.
And if you're building an app, you can "experiment with, preview, and submit your integration for approval with the Cards Validator."