Avatar: The Last Airbender: In the episode "The Storm", you see a younger girl standing close to Iroh and smiling as Zuko gets scarred in the Agni Kai. This lets you know two things: One, she's family, and two, she's gonna be trouble when she finally shows up. Ladies and gentlemen, you just met Princess Azula.
Azula is just one of many examples; the show loves this trope. The Giant Lion-Turtle who finally unlocks the Eleventh Hour Superpower for Aang was mentioned as early as the pilot, appeared several times as a statue, and showed up on a suspiciously conspicuous scroll in the ancient library.
In the second season of Archer, Cheryl mentioned in passing that she had brother named Cecil. Cecil finally appeared in the fourth season as a major character in the two-part Season Finale.
In the first season of Bob's Burgers, there was a brief scene where Tina mentioned having fantasies about Dr. Yap, the family's dentist. Dr. Yap finally appeared in the second season in a self-titled episode.
Batman: Mask of the Phantasm has a mobster that appears in the back of an old photo and is seen when Andrea and Bruce are about to talk to Andrea's father about the engagement. Turns out that's the man that would later become the Joker.
In Code Lyoko, William is introduced as the token "bad boy", played to be a romantic rival for Ulrich and not much else. However, in the third season finale, he's inducted into the Lyoko Warriors, gets possessed by XANA, and ends up being the enemy for the next season.
Codename: Kids Next Door: Leaky Leona appears briefly in the crowd of kids attending the DC's party in "Operation: Date" a whole season before getting her very own story ("Operation: Fountain"). The future fourth-grade president appears briefly in "Operation: Archive" trying to eat a candy bar and forgetting he's wearing a fishbowl helmet (he's on the moon).
A handful of episodes have a very wimpy adult man show up in flashback. Come Operation Z.E.R.O., we find out that he's Father without the shadow suit.
In Danny Phantom, a girl named Valerie appears a couple of times. She gets maybe four lines of dialogue at max and seems to be nothing more than a standard Rich Bitch who wouldn't give the main character any time of day. Then her "debut" episode came. From that point on, she becomes the series' Anti-Hero.
In Futurama, Leela's parents are in the background of a large crowd of Mutants in an episode some time before they were revealed in the plot, Nibbler's shadow appears in the first episode, and the Number 9 Guy from ITWGY appears at various points during the series...
The Number 9 Guy was actually planned to be important in a canned storyline before that major appearance.
Nibbler's shadow does not appear in the original pilot. It was added later. Sometimes the unedited version still airs.
Also, Amy Wong, Dr. Zoidberg, Hermes Conrad, Kif Kroker, Zapp Brannigan, Scruffy, and Smitty all flash past during the opening credits of the first episode. Smitty shows up with Url later in the episode, Amy, Zoidberg and Hermes don't show up until the second episode, Kif and Zapp don't show up until the fourth, and Scruffy doesn't appear until the sixth.
When Nibbler is introduced as a character, he just appears to be a pet for Leela and we're led to believe his eventual significance is to make fuel for the ship, as his bodily waste comes out as dark matter. But that's still not it. Not even close. He is an alien from a highly advanced ancient race that has been at war with the floating brains. And he was responsible for making Fry get frozen.
In Disney's Gargoyles, a lot of characters would be introduced like this. For example, Elisa Maza appears for a few seconds in the first two episodes before being formally introduced in the third. Matt Bluestone can be seen driving the police chief Maria Chavez around, one episode before we're actually told who he is. The Archmage at first seems like a one-time villain to use in a flashback, until he returns with a vengeance. The uber example of this, however, has to be Vinnie, who is occasionally seen in the background having bad things happen to him thanks to the gargoyles, until he is focused on in an episode where he decides to get revenge by shoving a pie into Goliath's face.
Titania is first introduced as Anastasia Renard, Fox's mother, several episodes before she appears on screen, where she comments that Goliath earned her favor due to previous services rendered. About two episodes later, Anastasia and Titania are revealed to be the same person. Similarly, in his debut episode, Puck mentions that he works for humans, one in particular ("the human" in Demona's words), because they are fun. Nearly 40 episodes later he is revealed to be Xanatos' right-hand man, Owen Burnett, who had been a recurring minor character since episode two. Word of God insists that many of these developed around mid-season 2, mostly after the writing staff realized how much stuff said offhand came together with the newer stories.
And you have Brendan and Margot, two recurring victims of events involving the gargoyles. Margot then later becomes adviser for the NYPD Gargoyle Task Force when the clan is revealed to the world, and she is an Assistant District Attorney in the Goliath Chronicles.
Mildly subverted in Harvey Birdman, Attorney at Law, with Bill Ken Sebben, Phil's twin brother. His introduction is so brief that it is even accompanied by the voice-over, "... and his twin brother, Bill, who was not worth mentioning up until this point." However, many episodes later he is mistaken for Phil, who died in the previous episode.
No, he didn't, everyone just thinks he did.
Jackie Chan Adventures lives off this trope. Villains who made first appearances in filler episodes of one season usually ended up becoming the Big Bad of later seasons.
Kim Possible has a few of these. When Ron Stoppable had a speech Dr. Vivian Porter was briefly seen in the audience; in the next episode she got a main supporting role. Also in that episode Justine Flanner was scene in another set of an audience, and she got a main supporting role several episodes later.
Warmonga appears in an early episode in the final season, and then becomes one of the main antagonists along with Warhawk in the Grand Finale, "Graduation".
The first episode of My Little Pony: Friendship Is Magic introduces Applejack's little sister Apple Bloom in passing during the extremely crowded Apple Clan reunion. Later on, there's a brief shot of her with two other fillies. Later in the season, Apple Bloom plays a key role in the events of "Bridle Gossip", and gets the spotlight in "Call of the Cutie", which also officially introduces the other two fillies, Scootaloo and Sweetie Belle, the latter of which is Rarity's little sister.
Big Macintosh and Granny Smith from the family reunion; the former seems well on his way to becoming a regular.
An episode-specific example: "A Friend in Deed" introduces a donkey named Matilda early on. She turns out to be Cranky's lost love.
Near the beginning of "Keep Calm and Flutter On", Discord casts a spell that causes a group of beavers to become mean-spirited and they later appear near the end and build a dam that causes Sweet Apple Acres to flood.
Averted in Twilights Kingdom Part 2, bordering on subverted for some viewers. Scorpan helped defeat Tirek in the first place, so one would expect Celestia to send Twilight to... Not look for him. Or even mention asking him for help at all. Odd.
A similar accidental example, though not in the same series, can be found in The Ren & Stimpy Show, which had a character named George Liquor, who had a starring role in the episodes "Man's Best Friend" (kept off of Nickelodeon) and "Dog Show", as well as some cameo appearances here and there. Nickelodeon didn't like the character, and gave the rights to him to John Kricfalusi once he is fired from the series' production. George Liquor has starred in his own webseries since then, which is currently getting a revival.
Franchise/Scooby-Doo has this happen all the time; some of the newer renditions try to lampshade or avert; but they'll still be surrounded by episodes that do it straight.
Sideshow Bob, who started out as simply being Krusty's assistant during Season 1 of the show, until a certain episode later in that season ("Krusty Gets Busted" Which featured Bob framing Krusty for robbing the Kwik-E-Mart in order to supplant the clown from his own show.) introduced the now-familiar mad criminal genius angle that's been Bob's forte from that point on.
Manjula first appeared as a little girl in Apu's flashback in the seventh season episode Much Apu About Nothing, in which Apu tells her that he is sorry that their arranged marriage will not happen, before he travels to the U.S. She comes back in The Two Mrs. Nahasapeemapetilons where Apu finds that he can't escape his arranged marriage with her.
South Park: In the Coon and Friends saga: who'd have thunk that Mint-Berry Crunch, who came out of nowhere, had no real significance to the plot, and was a noticeably lame superhero, would wind up being the one to fix everything? Even Cartman's surprised.
Cartman: (in disbelief) "Fucking Mint-Berry fucking Crunch..."
The frequent Early Bird Cameos in The Spectacular Spider Man function this way for a viewer unfamiliar with the Spider-Man mythos, as ostensibly tertiary characters are developed into supporting cast and antagonists.
Done to great effect in Spider-Man the Animated Series in the early 90's. Nearly every character who later becomes important as a villain or another hero appears first as an acquaintance of Parker. Dr. Connors, Felicia Hardy, etc. Eddie Brock in particular appears often in the first few episodes, and becomes Venom later in the season.
In Young Justice, the audience is introduced to Barbara, a minor student at Artemis' school who has only one line of dialogue in the entire first season. At the start of season 2, she's joined the main cast as Batgirl.
There's also Karen Beecher (Bumblebee), Mal Duncan (Guardian II), and La'gaan (Lagoon Boy), all bit characters and classmates of the team, with less than six lines in the previous season who joined the team for the second.
Porter C. Powell in Transformers Animated is an accidental example. He was twice in the first season: the first is as one of Prometheus Black's financiers letting him know that his funding is being cut and he's not getting anymore test subjects, the second is as a random bystander when Professor Sumdac unveiled the improved Dinobots. In the first episode of season two we find out that he's Chairman of the Board of Sumdac Systems and proceeds to take over the company in Sumdac's absence and kick Sari out of her home for using money to try and find her dad, solidifying his role as both a Corrupt Corporate Executive and the most important human villain. However, his role and his position on the board only came into being when the show's staff members took a liking to his character design and bugged the head story editor into featuring him in a larger role.
The Venture Bros. practically runs on this trope. At the end of season one, The Monarch's cocoon base is destroyed. It is rebuilt surprisingly quickly at the start of season two, and this is handwaved by having Monarch say he used parts he stole from an unknown character named Sargent Hatred. At the end of the season, Hatred makes an appearance at The Monarch's wedding, but it's essentially a one-off gag. Shortly after that, in the episode that reveals Billy Quizboy and Phantom Limb's shared backstory, the shouty, red-haired NCO who demotes Brock to Operation: Rusty's Blanket is shown putting on a Guild of Calamitous Intent signet ring, revealing that he is, in fact, Sergeant Hatred, before he got his distinctive tattoos. Then in season three, Hatred becomes Dr. Venture's new arch-enemy, and it turns out his entire motivation for doing this is to get back at the Monarch for stealing from him (by stealing Monarch's nemesis.) This leads directly to Hatred becoming the new Venture bodyguard and a main character in season four.
Then there's Captain Sunshine, who is given a passing mention in season one (Monarch tells his mooks to "send the charred remains of Wonder Boy to his beloved Captain Sunshine.") More than two seasons later, that line of dialog spawns an entire episode, in which we meet Captain Sunshine, and see that the loss of Wonder Boy has caused him all sorts of mental problems.
Phantom Limb started off as a throwaway character attending the Venture yard sale, but he ended up coming back and becoming season two's Big Bad.
In W.I.T.C.H., Elyon, who would eventually become the focus of a major part of the series is first introduced as a classmate of the main group, and is even introduces herself with another character that would remain a background character.
During the second season finale, we're introduced to a man that is a major part of the next story in the comics, but as the series didn't get another season, it's technically a Continuity Cameo.
In the Adventure Time episode "The Creeps", Finn gets freaked out by this terrifying ghost-creature. By the end of the episode he assumes it was a prank by his friends, but they swear they don't know what he's talking about, and its never explained. Turns out its a vision from a past life, trying to send him a message.
Also, the snail. Appears in every single episode as a "where's Waldo" type of deal. Thenhe gets possessed by the Big Bad and spends about two seasons trying to destroy all life.