The spider defeated by Samwise Gamgee in Tolkien’s The Two Towers is Shelob.
Shelob is a gigantic spider living in Cirith Ungol, a pass near Mordor. She attacks Frodo and Sam while they are trying to enter Sauron’s land. Her venom paralyzes Frodo, and the Orcs who find him assume he is dead.
Sam refuses to abandon his master. Using the Phial of Galadriel and the sword Sting, he confronts Shelob and wounds her badly enough to drive her away. He then takes the One Ring briefly, believing Frodo has been killed, before discovering that Frodo is unconscious rather than dead.
Shelob is not simply an ordinary oversized spider in Tolkien’s mythology. She is described as a descendant of Ungoliant, an ancient and monstrous being from the earlier history of Arda. The character’s encounter with Sam occurs near the end of The Two Towers, whose book structure differs from the film adaptation’s placement of events.