In Harry Potter and the Goblet of Fire, Harry falls into the Pensieve in Dumbledore's office and experiences Dumbledore's memory of Karkaroff's confessions.
During this memory, Karkaroff mentions Snape as a Death Eater, at which point Dumbledore comes to his defence.
Admittedly, I can't remember the books well enough, but does Harry know about Snape being a Death Eater at this point? In the films, I'm sure he doesn't.
If so, why does he not react to this news?
Harry did not react right away, as he was viewing the memory alone:
“Snape!” he shouted. “Severus Snape!”
“Snape has been cleared by this council,” said Crouch disdainfully. “He has been vouched for by Albus Dumbledore.”
“No!” shouted Karkaroff, straining at the chains that bound him to the chair. “I assure you! Severus Snape is a Death Eater!”
Dumbledore had gotten to his feet.
“I have given evidence already on this matter,” he said calmly. “Severus Snape was indeed a Death Eater. However, he rejoined our side before Lord Voldemort’s downfall and turned spy for us, at great personal risk. He is now no more a Death Eater than I am.”
Harry turned to look at Mad-Eye Moody. He was wearing a look of deep skepticism behind Dumbledore’s back.
There was too much going on in the memory to be distracted, probably Harry had suspected this turn of events. After viewing the Pensieve memories, having been interrupted by Dumbledore, Harry raises the question and gets an answer:
... “No more has Professor Snape,” he said.
Harry looked into Dumbledore’s light blue eyes, and the thing he really wanted to know spilled out of his mouth before he could stop it.
“What made you think he’d really stopped supporting Voldemort, Professor?”
Dumbledore held Harry’s gaze for a few seconds, and then said, “That, Harry, is a matter between Professor Snape and myself.”
Harry knew that the interview was over; Dumbledore did not look angry, yet there was a finality in his tone that told Harry it was time to go. He stood up, and so did Dumbledore.
Harry did not question authority of Dumbledore before the events of OotP. He was probably convinced by the answer he got.
As we have seen in both movies and novels, he has trusted Dumbledore very much. So what Dumbledore said after when Karkaroff said Snape is a death eater, he believed it and didn't question it.
"I have given evidence already on this matter," he said calmly. "Severus Snape was indeed a Death Eater. However, he rejoined our side before Lord Voldemort’s downfall and turned spy for us, at great personal risk. He is now no more a Death Eater than I am."
-Goblet of Fire, chapter 30 (The Pensieve)
And there were more things to worry about at that moment than this news. Bartemius Crouch was found murdered. Obviously, it was an important thing to take care of. Dumbledore was trying to find culprit behind all this by looking into old memories and link the broken chain of events.
Besides this, the most important thing was that Dumbledore trusted Snape and Harry trusted Dumbledore. Therefore, he didn't question trust of Dumbledore.
However, there were a few times when Harry doubted Snape, but he had to drop the topic each time.
In Philosopher Stone,
Hagrid: If Dumbledore trusts Snape, then so do I and then so should the three of you.
In Half Blood Prince,
Remus Lupin: If Dumbledore trusts Snape, then so should they.
Months before the incident with the Pensieve, Harry had a conversation with Moody (Crouch Jr.) about why Crouch Sr. would be searching Snape's office. During that conversation we find the following tidbit:
"Oh if there's one thing I hate," he muttered, more to himself than to Harry, and his magical eye was fixed on the left-hand corner of the map, "it's a Death Eater who walked free...."
Harry stared at him. Could Moody possibly mean what Harry thought he meant?
I think it is quite plausible that the thing that Harry wondered whether Moody could possibly mean was that Snape had been a Death Eater who walked free. As Moody is (in Harry's mind) a renowned Auror and close friend of Dumbledore, Harry would likely take this very seriously. If this was not enough to convince him that Snape had been a Death Eater, he would probably have been convinced by the conversation with Sirius in Hogsmeade:
"I think they've both got a point," said Sirius, looking thoughtfully at Ron and Hermione. "Ever since I found out Snape was teaching here, I've wondered why Dumbledore hired him. Snape's always been fascinated by the Dark Arts, he was famous for it at school. Slimy, oily, greasy-haired kid, he was." Sirius added, and Harry and Ron grinned at each other. "Snape knew more curses when he arrived at school than half the kids in seventh year, and he was part of a gang of Slytherins who nearly all turned out to be Death Eaters."
Sirius held up his fingers and began ticking off names.
"Rosier and Wilkes — they were both killed by Aurors the year before Voldemort fell. The Lestranges — they're a married couple — they're in Azkaban. Avery — from what I've heard he wormed his way out of trouble by saying he'd been acting under the Imperius Curse — he's still at large. But as far as I know, Snape was never even accused of being a Death Eater — not that that means much. Plenty of them were never caught. And Snape's certainly clever and cunning enough to keep himself out of trouble."
So the scene that Harry witnessed in the Pensieve may very well just have been the confirmation to what he had long suspected.
I've always wondered this too. Even when he asks Dumbledore it's more of an afterthought and he focuses more on why he turned good rather than the fact that Snape ever was a Death Eater. Remember though he and Sirius had already talked about this earlier. Sirius told him that Snape always had a talent/interest in Dark Magic and was friends with some of the more notorious Death Eaters, but he was unsure if Snape himself ever was a Death Eater. Also (fake) Moody was treating him the same way he treated Karkaroff, who Harry found out quickly was a Death Eater and who knew Snape pretty well. Harry probably already was fairly certain that Snape is/was on Voldemort's side and didn't think much of it when he was outright told.
