The official Twitter client. Honestly, if you're reading this question, you're probably looking for alternatives to this, but it's a good benchmark to measure other Twitter clients against.
Ranked in these QuestionsQuestion Ranking
Pro Gets the latest API updates
Twitter is slightly infamous for making changes to the API. At least with the official client, you stay ahead of this.
Pro App experience similar to web experience
If you're switching from using the web client, this app will be quickly understandable to you.
Pro Multiple account support
Allows easy switching of accounts withing the app.
Pro Supports login verification
The official Twitter app is perhaps the only app to support two factor authentication or login verification, thus providing a necessary additional layer of security to your Twitter accounts
Pro Photo editing tools
The app has filters, automatic level balancing and ability to crop built in the app.
Pro Clean & Functional UI Design
Pro Officially sponsored so you know it will always be supported and won't be randomly shut down
Pro Inline Photos
Con No widget
Recently took away the widget for whatever reason... and that's not cool.
Con Terrible cellular data management
If you have a limited data plan for your phone, don't be surprised if you find a whopping 800 MB consumption at the end of the month. The app really, really wants to download absolutely everything, even if you don't want to see or play anything.