Here's what happens:
* Wormtail uses the Killing Curse (Avada Kedavra) on Cedric, on Voldemort's orders.
* Wormtail was acting under Voldemort's control, and Voldemort wanted to use Cedric's death as a way to lure Harry Potter into a trap.
It's important to note that while wands are used in the wizarding world, the Killing Curse is a powerful magic that can be cast without a wand, as Wormtail demonstrates.