Can I use Airplay to run an app on my TV?

Not at the moment.Perhaps in a future update Apple will allow for apps to runn on the ATVs. Airplay mirroring will be coming to the MAc with the OS X10.8 mountain lion update though

Just a correction. The OP has an iPad 2 and that has the ability to be wirelessly mirrored to the ATV. The majority of apps do work.