What is a Tanin?
Rashi: Tanin means a snake. 1
Whose staff did Hashem instruct Aharon to take?
Why did it turn specifically into a snake?
Targum Yonasan: It was a prediction that the whole world were destined to hear the cries of the Egyptians when Hashem would break them, just as the whole world heard the screams of the snake (in Gan Eden) when Hashem cut off its hands and legs.
Da'as Zekeinim: Because Pharaoh was vain, and therefore became known as the great Tanin; and because, just as a snake twists and turns as it moves along, so too, did Pharaoh twist and turn. 1
Rosh (6:3): Pharaoh is called a snake - "ha'Tanin ha'Gadol." Dry wood became a snake, swallowed Matos (staffs) and reverted to dry wood. So I created you from nothing (Hadar Zekenim - from a putrid drop of semen), and made you king, and you said "the river is mine, and I made it"; you made yourself like a snake, and swallowed 12 Matos (Shevatim) - you will return and expel them, and I will return you to Tohu v'Vohu (nothingness).
See also Chizkuni and Ba'al ha'Turim.