ScoreCenter does exactly what you'd think it does: it gives score updates from a variety of sports. I have four sports I follow: (NFL & NCAA), baseball, and hockey. Though ESPN does a good job of including many, many more sports (here's a full list).
What's super cool about this app is that it lets you passively follow your favorite teams. Once you pick a sport to follow, you can pick your specific teams to follow. Those teams will be highlighted within the app for you. You can also set various push alerts for each team.
The alerts are the amazing part of this app for me. There are alerts for the start of the game, and various score updates. For the most part I just get alerts on the final score (see the Alert Details screen shot). The benefit of this is that it lets me keep tabs on my teams without having to continue to check in to ESPN all the time.
For all of the teams I follow, I have the final alert set. It tells me the score along with a couple highlights from the game. Sometimes, like during the NFL post season, I add alerts for each scoring play. That is really exciting!
As the myTeams screen shot shows, you can tap on each game and get all the details. I typically only care about the score, but it is nice to have all this data easily accessible.
This app would be amazing if it did what MLB's iPad did: which is show live games right on the device. I realize this is league dependent, but this would be huge! Image live streamed World Cup Soccer games.
As I said, this app is excellent. It's great being able to follow my teams and root for them throughout the day. I recommend checking it out.