United Airlines has one of the best apps on the App Store. It does all the basics, purchase, upgrade, check-in, boarding passes etc.
The real advantage for me is a feature that no other airline offers. When you look at flight status it allows you to check where the aircraft is coming from. You can use this to trace the plane back throughout the day and it helps anticipate delays. Because the airline doesn't "delay" a flight until its wheels up time is in jeopardy this is a big help. Also it allows you to set flight updates for the inbound flight so you can tell when your aircraft reaches the gate. This is extremely helpful if your waiting for the flight in the United Club or somewhere else in the airport.
On the lines of predicting delays it does not give you information on the status of the crew (in the event a crew is coming in on another aircraft). Overall this has made traveling on United more predictable and easier. Recently I've tried out apps from Delta, US Air, Southwest and Alaska. None of these carriers provide this type if data on a mobile app.