The Wizarding World of Harry Potter has an extensive history and fans learn about many important events that predate the main series. From the founding of Hogwarts to Harry's parents' time as students, a lot of this backstory is integral to the main plot of the movies.
That being said, there are a few occasions when the movies allow fans to take a glimpse back into the past, whether through the use of Tom Riddle's diary, the Pensieve, or dream sequences. While these are used sparingly throughout the series, several important characters' pasts are seen within flashbacks.
Igor Karkaroff
In the grand scheme of the series, Karkaroff is a fairly minor character and only appears in the fourth movie. He is the heaster of Durmstrang school and arrives when Hogwarts hosts the Triwizard tournament.
It soon becomes clear that Karkaroff has a prior connection to Snape which furthers the mystery of the Professor's true allegiances. In one scene, Harry uses the Pensieve and sees Karkaroff on trial for being a Death Eater and he reveals that Snape is also a former Death Eater in this scene. While fans don't discover that much about Karkaroff here, it does show just how cowardly he is as he is prepared to sell out his accomplices in order to lessen his own punishment.
Barty Crouch Jr.
The flashback of Karkaroff's trial also finally reveals the identity of the mysterious Death Eater Harry sees earlier in a dream, and then again casting the Dark Mark at the Quidditch World Cup. Karkaroff reveals that Barty Crouh Jr. is a Death Eater and Harry then realizes this is the same man.
Crouch's sudden switch in demeanor here is chilling and although the character is severely underused, this is a great introduction to the Death Eater. Harry learning who Crouch is and that he is in Azkaban is instrumental as it plays an important part in the later reveal. While it's odd that Crouch is never seen or mentioned again in later movies considering he is one of the most powerful Death Eaters in the series, this flashback is very much needed in order for the Polyjuice Potion reveal to work.
Albus Dumbledore
Dumbledore is one of the Harry Potter characters with the most screen time so it's unsurprising that he features in several flashbacks throughout the series. While young Dumbledore is the focus of the Fantastic Beasts movie, fans first got to see the character's past back in the second Harry Potter movie. Dumbledore's interactions with the young Tom Riddle in this flashback are especially important as they serve to show the parallels between Voldemort and Harry.
Later on in the series, the younger Dumbledore is shown again when he visits an even younger Riddle at his Orphanage. Despite the sheer brilliance and intelligence of his character, the flashback shows that Dumbledore fully believed he could set Riddle on the right path and steer him away from his darker tendencies (which had already begun to manifest). Most of Dumbledore's flashbacks show the many similarities between his prior relationship with Riddle and his current one with Harry.
Severus Snape
Snape is a constant thorn in Harry's side throughout the series and fans were made to question his loyalties from the get-go. The most important flashback sequence in the movies actually centers around Snape, though.
In the final movie, Snape is killed by Voldemort and Nagini and Harry takes his tears to the Pensieve in order to view his memories. Through this, the audience learns that Snape had been in love with Harry's mum, Lily, and only killed Dumbledore after Dumbledore asked him to. Snape's true character is only finally revealed to the audience within these flashbacks which inspires Harry to even name his son after him.
Rubeus Hagrid
Hagrid is the person that takes Harry into the Wizarding World and is the person to carry him out of the Forbidden Forest in the final movie. A younger Hagrid is seen when Harry is transported into the past via Tom Riddle's diary.
Viewers learn that Hagrid was expelled from Hogwarts as he was accused of opening the Chamber of Secrets. Although the scene mostly acts as a diversion, it is great to see the lovable Half-Giant get so much focus. It's also interesting to see why he got expelled after it was teased in the first movie.
Tom Riddle
Although Voldemort's younger self, Tom Riddle, is manifested by the diary Horcrux at the end of the second movie, this version of the character is mostly seen through flashbacks. When Harry learns the truth about Hagrid's expulsion from Hogwarts, Riddle is also present in these flashbacks.
While Harry does defeat the memory of Riddle in the second movie, he learns more about his past through the Pensieve in the sixth movie. Here, Harry sees Dumbledore visiting Riddle in his Orphanage and then sees him at Hogwarts asking Slughorn about Horcruxes. These memories/flashbacks lay the groundwork for Harry to defeat Voldemort as he is able to see his origins and learn more about Horcruxes.
Horace Slughorn
Slughorn is the new Potions Master in Harry's sixth year at Hogwarts and pretty much from the off, it's clear that Dumbledore has reinstated him for a much deeper reason. Dumbledore enlists Harry's help and instructs him to gain the new Professor's trust and it soon becomes apparent that he is keeping an important secret relating to Tom Riddle.
By the time the sixth movie came around, fans had learned to be distrustful of new teachers but Slughorn is actually one of the best Professors at Hogwarts. These scenes perfectly capture the complex nature of Slughorn's character; he 'collects' talented students during both his stints at Hogwarts so it makes sense that he wants to conceal his mistake, even if it puts the Wizarding World in greater danger.
Gellert Grindelwald
Before Voldemort rose to power, Grindelwald was another dark wizard threatening the Wizard World. He is the main antagonist of the Fantastic Beasts series but he also features briefly in the penultimate Harry Potter movie.
Voldemort visits the aging Grindelwald in prison but viewers also see a brief flashback to years prior when he stole the Elder Wand. Voldemort's quest for the Elder Wand is an important part of the final two movies so while this flashback is brief, it sets up a vital element of the story. It's also great to look back at this scene in retrospect as the same actor (Jamie Campbell-Bower) plays young Grindelwald here and in the Fantastic Beasts series.
Lily Potter
While he does share a lot of similarities with his father, fundamentally Harry is most like his mother, Lily. Prior to the final movie in the series, Lily is seen very briefly in flashbacks but fans don't get to learn that much about her character beyond her sacrifice.
It is through Snape's memories in the final movie that Lily's past is properly explored. Her strained relationship with her sister, her friendship with Snape, her Sorting, and the beginnings of her relationship with James are shown through these memories. Not only do these scenes reveal her prior connection to Snape, they also delve into the true power of her sacrifice and why Harry lived on that fateful night when Voldemort tried to kill him.
James Potter
Flashbacks to the night of Harry's parents' deaths are seen as early as the first movie when Hagrid tells him about Voldemort. Although his death is what started the whole series, the most revealing flashbacks about James actually come in the fifth movie when Harry infiltrates Snape's mind and sees his past.
Prior to this, Snape makes no secret of his disdain for Harry's father but the stern potions master doesn't feel like the most reliable source. Harry sees that his father and his friends made Snape's life a misery when they were at Hogwarts; these scenes make Harry question everything he thought he knew about James and tie into his ongoing internal battle with Voldemort. The scenes also cleverly foreshadow the reveal of Snape's feelings for Lily and his true allegiances at the end of the series.
