It's no secret that when J K Rowling was working on the Harry Potter series, she planned things out extremely carefully, laying the groundwork for future reveals from the very first pages of Harry Potter And The Philosopher's Stone. Throughout the first book, there are layers of meaning - like in Snape's potion class, when he asks Harry about specific ingredients - and moments of foreshadowing.
However, when the book was adapted to film, quite a bit of it was cut. For the most part, this was nothing major - filler scenes and extra exposition. However, some instances of carefully planned foreshadowing and build-up were cut as well.
Dumbledore's Love Of Lemon Drops
In the opening scenes, when Dumbledore and McGonagall are waiting for Hagrid to come to Privet Dr with baby Harry, Dumbledore offers her a lemon drop. Of course, this foreshadows the passwords to his office, which are all the names of muggle sweets, and which become significantly more important later on.
However, this is a very minor omission. It's not too important that Dumbledore previously expresses a like for the sweets, and it's easy to see why it was cut. Given that this is far from the only thing that would be cut from the opening scenes, it really has no impact on the film overall.
Dumbledore's Watch
In the same scene that Dumbledore expresses a love of muggle sweets, he is also described as checking his watch, which "had twelve hands, but no numbers, with little planets moving around the edge". This is later revealed to be, presumably, a watch given to Dumbledore when he came of age - as Molly gives one to Harry, and explains that a gift of a gold watch is a wizarding tradition.
Like his love of Lemon Drops, though, this is a tiny detail that would easily be left out of the film with no major repercussions - especially as the first film was made in 2001, and the scene where this tradition was explained wasn't until the Deathly Hallows - which was released as a book in 2007! There was no way to know that this was anything but a casual detail to make things more magical.
Marge Hates Harry, & Mrs. Figg
It's mentioned in the book that Marge, Harry's aunt, "hates" Harry, and that Mrs. Figg often looks after him. This comes up because Harry wasn't originally meant to go to the zoo, and the Dursleys are trying to think of another person to care for him. Of course, both characters become important later - Marge is so rude that Harry blows her up and runs away, and Mrs. Figg is revealed as a member of the Order of the Phoenix!
However, the entire exchange is left out of the movie- Harry was always intended to go to the zoo, and neither Marge nor Mrs. Figg are mentioned at this point. Once again, though, this is a minor omission. Both do still play their key roles later on, and while a little depth is lost here, it's not enough to really impact the story.
Descriptions Of Lily & James' Wands
The scene where Harry gets his wand at Ollivander's goes through some changes, too. One major one is that Ollivander of the books tells Harry about his parents' wands. Lily's, he says, was "a nice wand for charm work", and James' as excellent for Transfiguration". Later, Slughorn talks at length about Lily's charm work, and the charmed goldfish she created for him, and James is later revealed to have mastered the most impressive transfiguration of all, and become an Animagus as a teenager.
Clearly, Rowling was setting up this particular skill from the start, and having it repeated by multiple characters solidifies Lily's character and adds to the reveal that James was an Animagus. However, it's easy to see why this might be cut, as it isn't entirely relevant, and detracts from Harry's magical wand-choosing experience.
The Centaurs
Harry first meets a centaur in both versions of the story, but in the book, several appear. Whereas film Harry is protected by Firenze, book Harry also meets 'villain' Bane and Ronan, who argue with Firenze about helping him. Firenze even says "I set myself against what is lurking in this forest, Bane, yes, with humans alongside me if I must". This foreshadows when Firenze is eventually exiled from the herd for helping humans, and comes to teach at Hogwarts.
However, in the film, none of this happens - and because Firenze doesn't teach at Hogwarts, this line setting up his exile isn't necessary. The story would have added depth to the storyline with Umbridge and the Centaurs, but in the end, it was presumably cut down so far out of necessity to keep the films at a reasonable length.
Neville's Broken Wrist
In both the book and film, Neville falls off his broom in class, and breaks his wrist - leading Madam Hooch to leave with him, Harry to fly after the Remembrall, and get picked for the Quidditch team. In the film, that's the last heard of Neville's injury, but in the book, Harry meets him later, and is told that the nurse fixed it "in about a minute". This comes into play later, when Harry is injured, and ends up having to regrow bones, because Professor Lockhart tried to fix it!
This isn't a major change, but it is interesting to realize that Harry, and everyone else in his class, knew without a doubt how easy it was for Madam Pomfrey to fix bones. This then adds to the distrust and dislike of Lockhart, as they would have known just how badly he messed up the spell.
How Old Scabbers Is
In both the book and the film, Ron introduces Harry to Scabbers, and makes a comment about how unimpressed he is with his pet. However, in the book, he is talking about getting hand-me-downs, and says explicitly that Scabbers is an 'old' rat that was given to him by his older brother.
Of course, this is a huge plot point later on, as it's revealed that Scabbers is Peter Pettigrew in his Animagus form! In the film, there is no hint that Scabbers is any other than a normal rat, which actually lessens the impact of the moment when the truth comes to light.
Flamel On The Chocolate Frog Card
In the books, Harry first hears the name of Nicholas Flamel when he's reading the back of his first Chocolate Frog card. The Dumbledore card includes the notes that he is known for his work with the uses of dragon's blood, and "his work on alchemy with his partner, Nicholas Flamel". Harry later knows he has heard the name somewhere, but can't place it - until he eats another Chocolate Frog and remembers. In the movie, Hermione finds the same info in a book.
This is a much bigger change, and one that really puts the emphasis on Hermione, and her ability to figure things out. This is an understandable change, in many ways - it makes Hermione the hero, and ensures that no fan wonders why Harry wouldn't remember this in the first place.
The Motorbike Belonging To Sirius
In the opening scene, Hagrid flies down on a giant motorbike, carrying Harry. In the book, Hagrid explains that he borrowed the bike from Sirius Black, and when he leaves, he says that he should be getting it back to him. This, of course, foreshadows Black's role - and the big reveal in The Prisoner of Azkaban that Sirius was actually a friend to the Potters!
This may have been left out specifically to enhance the shock of that reveal - and, of course, because it doesn't initially matter how Hagrid got the bike.
Dragons At Gringotts (& Hagrid Wants One)
One of the key plot points in the first installment in the series is that Hagrid has set Fluffy, his three-headed dog, to guard the Stone... and that he accidentally tells someone how to get past Fluffy when convincing the stranger to give him a dragon egg. He's always wanted a dragon, and Harry eventually figures out that the one he gets was a setup.
In both book and movie, Harry tells Hermione and Ron that Hagrid has always wanted a dragon, and that he told Harry as much when they first met. In the book, this is true - Hagrid mentions that there are said to be dragons at Gringotts, and that he's wanted one since he was a kid. In the film, however, Hagrid never actually tells Harry this at all! This was an oversight, as a deleted scene shows the two on the tube, where Hagrid says he wants a dragon. Obviously, this was intended to be a part of the final film, but when it was removed, the plot hole this causes wasn't resolved!