In 1984's Terminator, the titular character comes from the future to kill Sarah Conner before she could give birth to John Conner. My question is what would happen if the Terminator managed to kill Sarah before giving birth to John?
The answer depends on how you view time travel in the Terminator universe.
By itself, the first movie presents a situation where there is only one timeline that can't be altered. It's not a paradox that John's father was sent back in time by John, because there was never a timeline where that didn't happen. Kyle Reese was always sent back.
If you take this view, then the Terminator couldn't have killed Sarah, because that's not what happened. If he had, then John wouldn't have been born, so Kyle wouldn't have been sent back, so John wouldn't have been born, so Kyle wouldn't have been sent back, so... MALFUNCTION.
Later movies and series make it clear that one can change the future by going to the past and creating a new, alternate timeline. Viewed this way, the first John Conner that sent back Kyle had a different father. He couldn't have had Kyle as his father, because he hadn't been sent back yet.
In this case, the terminator could have succeeded in his mission, killing Sarah. This would have meant no John Conner in the new future. Skynet's hope was that there would be no new leader to replace him, so that the machines would win the war. There's no way to know if this would have worked out, or if the pressures of history would have found someone new to lead the resistance.