How do I defeat Piranha Plants?

There are at least two ways to defeat the Piranha Plants.

1. As Adeese stated, if you can pick up an invincibility star you can just walk or jump straight through them.
2. Most levels with Piranha Plants also contain enemies with shells, i.e. Koopa Troopas or Buzzy Beetles. If you jump, stomp or spin strike onto these, this will send the shell off along the ground. The shell can then also kill the Piranha Plants. Of course, not every plant can be killed this way, but a lot of them are placed such that this is possible (particularly those that need to be killed to safely reach certain areas of a level).