The serpent that gnaws at Yggdrasil's root is Nidhogg, also written Níðhöggr or Nidhǫggr.
In Norse cosmology, Yggdrasil has three great roots extending toward different regions of the cosmos. Nidhogg gnaws the root associated with Niflheim, beneath which lies the spring Hvergelmir. The creature's constant biting is one of several forms of damage suffered by the world tree.
Nidhogg is described as a serpent or dragon in the Prose Edda and appears in the poems Grímnismál and Völuspá. It is also associated with the dead and with Náströnd, a grim region connected with punishment. Ratatoskr, the squirrel running through Yggdrasil, carries hostile messages between Nidhogg below and an eagle above.
Nidhogg is often confused with Jörmungandr, the Midgard Serpent, or Fafnir. Those are separate mythological beings: Jörmungandr encircles the world, Fafnir is a dragon from a heroic legend, and Nidhogg belongs specifically to Yggdrasil's roots.