"Jailbreaking" refers to unauthorized modification to the phone that allows it to install unapproved apps not from the App store. It voids the warranty, and creates various problems, including susceptibility to a new worm that steals data.
You should restore the phone as a new iPhone, not using a backup which may contain corrupted code causing this. After restoring, sync the phone to get your Apps. If the apps still don't open, there may be a hardware issue which needs evaluation at an Apple Store Genius Bar.