While the Hydra seen in the series shows the capacity to change shape, the average appearance of the species seems to be that of a large snake-like creature with multiple branching heads.
Despite their power, they are capable of being subdued, sealed and presumably killed, although this was admittedly not the case with Hachirotaro Okami, to Yukio's surprise.
Powers and Abilities
It has incredible divine power and regenerative ability, even when sealed, and a range of transformation abilities. It can also take control of anyone in its sight.
The Lernaean Hydra was a giant serpent in Greek and Roman myth. Defeated by the legendary Heracles, it became a popular choice for media, many such creatures cropping up in literature, sculpture, and other artwork.
The manga's Hydra also take inspiration from the Japanese Yamata no Orochi, a giant eight-headed snake. It also has the same kanji writing.