In Harry Potter and the Philosopher's Stone, Harry and his parents were attacked by Voldemort and Harry's parents died. But Harry survived because his mother had died to save him and as a result Voldemort's killing curse had rebounded upon him. But that is weird because if Harry is the only one to have survived the killing curse then Voldemort should have been dead. This leads me to think that Voldemort was able to survive the killing curse as well and so Harry wasn't the only one to survive it.
The Dark Lord did indeed survive it as well.
It is indeed correct that the Dark Lord survived the Killing Curse as well. He mentioned this himself when addressing the gathering of Death Eaters in the graveyard.
“You know my goal – to conquer death. And now, I was tested, and it appeared that one or more of my experiments had worked … for I had not been killed, though the curse should have done it.”
- Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)
Though it remains a rare occurrence, it is true that Harry was not the only one in history to survive the Killing Curse. The Dark Lord was able to survive the Killing Curse because he had Horcruxes, which tethered his soul to earth so he could not truly die even if his body was destroyed.
“Well, you split your soul, you see,’ said Slughorn, ‘and hide part of it in an object outside the body. Then, even if one’s body is attacked or destroyed, one cannot die, for part of the soul remains earthbound and undamaged. But, of course, existence in such a form …’
Slughorn’s face crumpled and Harry found himself remembering words he had heard nearly two years before.
‘I was ripped from my body, I was less than spirit, less than the meanest ghost … but still, I was alive.’
‘… few would want it, Tom, very few. Death would be preferable.”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
Wizards who have created Horcruxes are themselves very rare, so it would be unlikely that many others have survived the Killing Curse in this way, but the fact remains that the Dark Lord did.
I think that when you say that:
"Harry is the only one to have survived the killing curse,"
you have in mind this quote said by the fake Mad-eye Moody in Harry Potter and the Goblet of Fire.:
"There is only one person known to have survived it (the killing curse). And he's sitting in this very room."
But I would like to draw your attention to some points to show that what Barty Crouch Jr. (the fake Mad-eye Moody) may have said was not entirely true.
After the mishap of the Godric's Hollow, no one knew what actually happened to Voldemort. Voldemort certainly had vanished but none had a clue if it was due to the rebounding killing curse or something else. Now when Barty Crouch Jr., a devoted death-eater, found his master, he most probably did not ask him about that night (he was a true servant and would never ever want his master to relive that horrific night).
There can be another explanation. Even if Barty Crouch Jr. came to know how his master disappeared, why would he carelessly tell this to anyone and attract attention to the fact that how much he knew about what happened that night (especially when his master was weak and needed his support in complete isolation from the wizarding world).
So, you are right in thinking that Voldemort too survived the killing curse (because of his horcruxes) and what the fake Mad-eye Moody said is not right.
It may not have been known that the curse rebounded on Voldemort
In Chapter One of Philosopher's Stone McGonagall says the following to Dumbledore:
No one knows why, or how, but they're saying that when he couldn't kill Harry Potter, Voldemort's power somehow broke — and that's why he's gone."
It seems clear that she does not know that the curse rebounded on Voldemort. Moreover, Dumbledore doesn't tell her that that's what happened, which would likely indicate that either Dumbledore didn't know either, or he wanted it to be kept secret.
Three chapters later we find the following exchange between Harry and Hagrid:
“But what happened to Vol-, sorry — I mean, You-Know-Who?”
“Good question, Harry. Disappeared. Vanished. Same night he tried ter kill you. Makes yeh even more famous. That’s the biggest myst’ry, see... he was gettin’ more an’ more powerful — why’d he go?
Notice that Hagrid doesn't give any information about what caused Voldemort to disappear. Again, it seems that average members of the Wizarding community such as Hagrid were simply unaware of what actually happened.
Thus, the claim that Harry was the only person to survive the Killing Curse should not be taken as evidence that Voldemort did not survive. It may simply be that those promulgating this fact were unaware that Voldemort had been hit by a Killing Curse.
I have to disagree with the submitter's understanding. When the curse rebounded, Voldemort's body died, and what was left of his soul split in two, one piece latching onto Harry, the other wandering in the wilderness (until Quirrell came along). To my mind, that doesn't entirely count as 'surviving' in any reasonable sense, as contrasted with baby Harry, whose body and soul were still alive and undamaged (save for the scar).