Ray Bradbury's "A Sound of Thunder" (the short story, not the less than mediocre film) is the paradigmatic case here, wherein a dinosaur hunter (another familiar time travel trope) steps off a marked path and squashes a butterfly. The upshot of that seemingly insignificant change in the past is that the future somehow feels different, as Bradbury so wonderfully describes:
Eckels stood smelling of the air, and there was a thing to the air, a chemical taint so subtle, so slight, that only a faint cry of his subliminal senses warned him it was there. The colors, white, gray, blue, orange, in the wall, in the furniture, in the sky beyond the window, were... were.... And there was a feel. His flesh twitched. His hands twitched. He stood drinking the oddness with the pores of his body. Somewhere, someone must have been screaming one of those whistles that only a dog can hear. His body screamed silence in return. Beyond this room, beyond this wall, beyond this man who was not quite the same man seated at this desk that was not quite the same desk... lay an entire world of streets and people. What sort of world it was now, there was no telling. He could feel them moving there, beyond the walls, almost, like so many chess pieces blown in a dry wind.Furthermore, the words on the wallsigns are spelled differently, and the wrong guy has just won the election!
As much as I love this story, it just doesn't add up. In increasing degrees of incoherence...
(1) While I buy the idea that a small change can have a dramatic effect (and this is where we get the phrase "the butterfly effect," after all), it's extremely unlikely that the ripple effect would leave the present intact enough for the same two candidates to be running for office but different enough for loser and winner to switch places. Likewise, while a change in language is possible, it's likely the change wouldn't simply be a matter of different spelling.
(2) The idea that the time travel agency could safeguard the timeline by creating a path and marking the target animals doesn't sit well with the idea that any minuscule change might be catastrophic. Surely they can't be tracking insects--and even microsopic organisms--to make sure none are inadvertently extinguished by a time traveler. Given the butterfly effect, it seems inevitable that any travel to the past would have serious repercussions in the present
(3) The past is the past (pace William Faulkner). It's already happened. So if anyone leaves the future in a time machine, their arrival in the past has already transpired and whatever they do has already been done. It isn't as if there is a pre-time-travel past (with a living butterfly) and a post-time-travel past (with a dead butterfly). No, there's just one past. So while time travelers may be able to visit the past, and if so they can certainly affect it, they cannot change it. Wanna know what that kind of time travel looks like? Watch 12 Monkeys.If I'm right about this, then many time travel stories don't make sense, in spite of how much fun they might be. This post has to end, but I can't help observing that Back to the Future's idea that as the past changes people will fade out of a photo from the future is especially ridiculous.
Chaospet (the source of the last link) has a nice trilogy of webcomics on this issue: 1, 2, 3.
Wondering about the possibility of multiple timelines? Stay tuned.