The Old World Wyrm is one of three confirmed dragon species in the Spiderwick canon (the other two being the Hydra and the Lesser Spotted Wyvern). It appears in Arthur Spiderwick's Guide to the Fantastical World Around You as well as the fantasy book series The Spiderwick Chronicles (specifically the final book, the Wrath of Mulgarath). Old World Wyrms are extremely venomous and poisonous: even touching one will cause a painful rash. They have many legs.
The story in the Spiderwick Chronicles involves an ogre, Mulgarath, stealing the field guide to fantastical animals to use the knowledge for evil purposes. The Field Guide mentions dragons grow faster when they drink cow's milk - something which some European myths allude to - and so Mulgarath starts to breed Old World Wyrms for his army, speeding their growth using cow's milk. The toxic aura of these dragons poisons the local water supply, which alerts the main characters of the book that something is wrong. In the conclusion of the book, the protagonists have to fight Mulgarath's army of goblins and dragons, battling against an adult Old World Wyrm using a Griffin. Once the Griffin vanquishes the dragon, the children then kill the infant dragons to prevent a dragon infestation.
The wyrm species is also mentioned in a companion book called Notebook for Fantastical Observations - the book is an interactive activity book which invites the reader to write in their own Fantastical Animal sightings alongside short stories from other people who have encountered the faery world. One of these stories involves a pair of children capturing a wyrm, mistaking it for a newt.
More can be found on the Spiderwick Wiki's page about dragons.