Short answers, pulled from the story.
Tolkien never settled definitively on which two towers the title refers to. He considered three possible pairs: Orthanc and Barad-dur, Minas Tirith and Barad-dur, and Orthanc and the Tower of Cirith Ungol. A cover illustration he drew named Minas Morgul and Orthanc as the pair.
The Two Towers was first published in 1954. It is the second of three volumes of The Lord of the Rings, preceded by The Fellowship of the Ring and followed by The Return of the King.
The Two Towers covers Books Three and Four of The Lord of the Rings. Book Three follows Aragorn, Gandalf, and the war in Rohan, while Book Four follows Frodo and Sam's journey toward Mordor. The narrative is interlaced, with each book setting aside the other storyline entirely.
Critics largely welcomed The Two Towers. Donald Barr in The New York Times called it "an extraordinary work." The Times Literary Supplement described it as a "prose epic in praise of courage."
Peter Jackson directed The Lord of the Rings: The Two Towers, released in 2002. The film was acclaimed by critics and was highly successful at the box office. Jackson added new scenes, imported material from the first film, and deferred the Shelob encounter to the following film.
A palantir is a seeing-stone Saruman used to communicate with Sauron. In The Two Towers, Wormtongue throws one from the tower of Orthanc. Pippin steals it from Gandalf and gazes into it, allowing Sauron to see him and mistake him for the holder of the One Ring.