If someone is put under an Imperius Curse, are they unaware for the duration - essentially unconscious while their body is under another's control? Or can they possibly remain conscious, able to experience what is happening, but unable to do anything whatsoever about it - a passenger in their own body?
Barty Crouch Jr, who has a lot of experience in both ends of the Imperius curse, tells us that its victims can be aware of their situation, although normally they are not:
‘But your father escaped,’ said Dumbledore.
‘Yes. After a while he began to fight the Imperius curse just as I had done. There were periods where he knew what was happening. My master decided it was no longer safe for my father to leave the house. He forced him to send letters to the Ministry instead. (…)’
(Goblet of Fire, chapter Veritaserum)
Harry, who is particularly talented at fighting off the curse, remains aware of his surroundings, but only barely:
It was the most wonderful feeling. Harry felt a floating sensation as every thought and worry in his head was wiped gently away, leaving nothing but a vague, untraceable happiness. He stood there feeling immensely relaxed, only dimly aware of everyone watching him.
And then he heard Mad-Eye Moody’s voice, echoing in some distant chamber of his empty brain: Jump onto the desk … jump onto the desk …
(…) suddenly Harry felt the empty, echoing feeling in his head disappear. He remembered exactly what was happening, and the pain in his knees seemed to double.
(Goblet of Fire, chapter Beauxbatons and Durmstrang)
So it is possible, but not what usually happens when someone is under the Imperius curse.
No, when the Imperius Curse is working, the Imperiused person won’t remember anything.
Once the Imperius Curse works properly and the target was not able to resist it, they do not have any memory of what happened while they were Imperiused. Katie Bell was successfully Imperiused to deliver a cursed necklace to Hogwarts, and she did not remember anything between after she opened the door to the bathroom where she was Imperiused until being in St. Mungo’s hospital.
“… that necklace … can you remember who gave it to you now?’
‘No,’ said Katie, shaking her head ruefully. ‘Everyone’s been asking me, but I haven’t got a clue. The last thing I remember was walking into the ladies’ in the Three Broomsticks.’
‘You definitely went into the bathroom, then?’ said Hermione.
‘Well, I know I pushed open the door,’ said Katie, ‘so I suppose whoever Imperiused me was standing just behind it. After that, my memory’s a blank until about two weeks ago in St Mungo’s.”
- Harry Potter and the Half-Blood Prince, Chapter 24 (Sectumsempra)
Therefore, the successfully Imperiused do not remember what happened while they were Imperiused.
Awareness is a sign of resisting the Imperius Curse.
Someone under the Imperius Curse isn’t able to resist is largely unaware of their actions, but Imperiused people can become aware if they are beginning to resist it. In all the situations we see someone described as becoming aware of their actions under the Imperius Curse, they’re beginning to resist it.
Both Barty Crouch Jr. and Barty Crouch Sr. start getting flashes of awareness when they start to resist the Imperius Curse after being under it for a long time. Barty Crouch Jr. describes one of the periods he becomes aware while resisting his father’s Imperius Curse as waking from a deep sleep, and then goes on to say he “found himself” out in public at the Quidditch World Cup, implying he might not have been aware of how he got there.
“But Winky didn’t know that I was growing stronger. I was starting to fight my father’s Imperius Curse. There were times when I was almost myself again. There were brief periods when I seemed outside his control. It happened, there, in the Top Box. It was like waking from a deep sleep. I found myself out in public, in the middle of the match, and I saw a wand sticking out of a boy’s pocket in front of me. I had not been allowed a wand since before Azkaban. I stole it. Winky didn’t know. Winky is frightened of heights. She had her face hidden.”
- Harry Potter and the Goblet of Fire, Chapter 35 (Veritaserum)
Harry is dimly aware of what’s happening the first time it’s used on him, but even that time, he’s trying to resist it, and partially succeeded in doing so. Harry’s experience therefore cannot be considered a typical experience of being controlled by the Imperius Curse, since he’s resisting it,
“He had both jumped and tried to prevent himself from jumping – the result was that he’d smashed headlong into the desk, knocking it over, and, by the feeling in his legs, fractured both his kneecaps.
‘Now, that’s more like it!’ growled Moody’s voice, and suddenly Harry felt the empty, echoing feeling in his head disappear. He remembered exactly what was happening, and the pain in his knees seemed to double.
‘Look at that, you lot … Potter fought! He fought it, and he damn near beat it! We’ll try that again, Potter, and the rest of you, pay attention – watch his eyes, that’s where you see it – very good, Potter, very good indeed! They’ll have trouble controlling you!”
- Harry Potter and the Goblet of Fire, Chapter 14 (The Unforgivable Curses)
From what we see, it seems like awareness goes hand in hand with beginning to fight off the effects of the Imperius Curse, and an Imperius Curse at its most effective leaves the victim with just a relaxed feeling, unaware of their actions. This would be logical, as if the victim knows they’re being controlled (especially if they’re aware they’re being made to do things they wouldn’t under their own free will) they’d be more motivated to fight against its effects.
I read the other answers and want to give examples from the movies
When moody shows the effects of Unforgivable curses on spiders, he brings one of the spiders near a water bottle to drown it. The spider is very afraid of the water and knows what is going to happen to itself, moves it's legs in panic to avoid the undeniable future. I think this part is also present in the books.
Second example is in Gringotts. When the trio go for Bellatrix's safe, Harry uses Imperius on one of the goblins at Gringotts. The face of the goblin suddenly becomes relaxed, and has no idea what is going on while under the effect of the curse. After the curse is broken, the goblin is very surprised to find himself near a dragon. But Harry is not a experienced Imperio caster and as others stated awareness only visible in terms of resistance and probably goblins do not have enough magical power to resist an Imperius curse.