Because "A Sound of Thunder" is an exploration and critique of the hubris of using advanced technology for frivolous purposes, the story has two settings: the age of dinosaurs and the "present day" in the United States in a near future shortly after a presidential election.
Eckels, the main character, has paid a large sum of money to travel back on a "safari" to kill a Tyrannosaurus rex. The present day of the story, as we encounter it through Eckels, is bureaucratic and technological, dominated by his experience at the time machine company. He and the man booking his safari are both grateful that Keith, rather than Deutscher, a potential dictator, won the recent election.
Bradbury spends more time on the dinosaur age, describing some of what Eckels sees as follows:
The jungle was high and the jungle was broad and the jungle was the entire world forever and forever. Sounds like music and sounds like flying tents filled the sky, and those were pterodactyls soaring with cavernous gray wings
Most of Bradbury's descriptive energy, however, is spent on describing the fearsome Tyrannosaurus rex—so much bigger than any animal in the present-day world—and the theoretical wonders and dangers of time travel.
Eckels and his party return from the safari, Eckels having caused problems by panicking and killing a prehistoric butterfly. Now, changes in the setting reveal that he has, in fact, changed history. He is first alerted to the change by spelling variations on the safari sign, then discovers that in this new version of the present, Deutscher has become president.