When things petrify, they turn into stone. Here's a piece of petrified wood, found in a forest in Arizona. Over millions of years, the wood slowly petrifies, or turns into stone:
When something petrifies you, it scares you so badly that you can't even move, as if you've turned into stone. If you're petrified of something, you're really, really afraid of it. And something petrifying is really, really scary.
Hermione points her wand at Neville and says the magic spell Petrificus Totalus. She petrifies Neville: turns him into stone.
Here, SpongeBob is petrified! He's so scared of something that he can barely move.