In Adventure Time it is never fully stated what happened to Finn's (human) parents. However, in the Adventure Time Encyclopaedia book it is revealed that Finn's parents abandoned him in a basket in the woods for unknown reasons. He was left there until he was found by Jake's parents. (The aftermath of the abandonment is seen in the flashback scenes of "Memories of Boom Boom Mountain".)
In Avatar: The Last Airbender, a large amount of supplementary information such as the history and details of locations or the names and background info of minor characters and animals comes from the Nickelodeon website and DVD Commentary. The creators have since expanded on this and additional info can also be found within the four Lost Scrolls books, based on screenplays of the show.
The second All-Avatar Nick Mag, in itself a collaboration of writers of the show and acclaimed comic artists, which contains comics that serve as a bridge for the time jump that occurs between the second Season Finale and the season three premiere.
As of 2010, there's also an artbook, which shows, among other things, the evolution of the character designs, background art, storyboards, sketches, and a ton of other stuff, including a little bit more in-universe background info.
The sequel series The Legend of Korra features only a brief tease about the dangling plot thread of Zuko's mother. You have to read the comics to find what really happened.
Quite a few background details for this series are also found primarily online; for example, the web serial "Republic City Hustle", as well as in the artbooks (such as where Raava and Vaatu came from).
A lot of character and setting details for Transformers are only found with the back-of-the-box toy descriptions for the characters and profiles released as supplementary material, occasionally with characters who never even made it into the show itself.
G.I. Joe also had a bit of this as well, one example would be the Dreadlok Buzzer whom was a disgruntled former Sociology professor in Cambridge England. Before he joined the Dreadnoks Richard "Buzzer" Blinken was a sociology college professor in England whom got in trouble because the college he worked at did not like his "extremist left-wing political beliefs". Buzzer wanted to do some research on Australian Biker gangs in which he ended up being a part of the very thing he was researching when he joined the Dreadnoks.
It should be noted that the box bios often have their own canon and storyline that might match the show's events, but for the most part writes it's own story. This is especially obvious in the Beast Wars toys. Since only 20 characters or so were introduced in the series, and dozens were created for the toyline, the differences are expected. Also, in the beginning the Beast Wars were set to take place after the Autobot-Decepticon war, among the humans. The early toys reference this, and the characters who appear in the show often have an entirely different characterization in their initial bios.
The Transformers Animatedwriting and art staff actually sat down and wrote a pair of manuals. The Allspark Almanac I and II are a pair of incredibly detailed books about the characters, setting, and plot devices in the show, including a lot of things that most cartoon writers would never think about in the first place. It's also ridiculously geeky.
The Total Drama Island Interactive flash site on Cartoon Network has bios for all the 22 campers for the show. This has very useful information for campers that got voted off early like Ezekiel, Eva and Noah, who got very little character development.
It's even useful when it comes to the main characters, for example, it reveals that Cody and Lindsay are both Spoiled Sweet, even though the series itself never mentions how rich they are.
While it's said in the show it's heavily implied that Lindsay was rich.
Each episode of Star Wars: The Clone Wars is accompanied by a comic (Season 1 only), episode guide, and creator commentary on the official site. While these normally just contain trivial info, they were all but necessary to watch the "Dooku captured" episodes. The comics explained Anakin's voluntary capture and showed how Anakin and Obi-wan got drugged, the episode guide explained how Dooku lost his sabers (the monkey-lizard pickpocketed him!) and the commentary video was Filoni explaining that he believed even Dooku can be captured by pirates if he's sufficiently outnumbered. These were also necessary for determining the chronological order of some of the earlier episodes. Later episodes in the series would normally take measures to avert this by keeping the timeline moving forward.
When Lilo & Stitch: The Series was still in production, Jess Winfield kept in touch with the fanbase at TV.com and helped confirm numbers, names, and functions of each experiment featured on the show. With Leroy & Stitch, all we need to find out are the remaining functions. Oh, and among other things, he confirmed that the 628 pod seen at the end of Experiment 627's episode was just a throwaway joke.
The Fairly Oddparents had a special "77 Secrets of the Fairly Oddparents" revealed Timmy's middle name (Tiberius) and Wanda's full name (Wanda Venus Fairywinkle). Also, a TurboNick special revealed Cosmo and Wanda's family tree.
Similarly, the Nick.com e-cards for ChalkZone told us that Rudy's full name is Rudolph Bartholomew Tabootie.
Dwayne McDuffie has a Q and A for Ben 10 on his site which details lots of info such as Ultimate Humungousaur can't grow, Way Big is bigger than Ultimate Humugousaur and that Azmuth didn't make the ultimates, Albedo did.
The first series had a marathon with "Omnifacts", which gave an extensive, off-screen backstory for Ghostfreak, explaining how Vilgax survived the first season finale, as well as hinting at characters in future series such as Seven-Seven, the Live-Action Adaptation also had Omnifacts, which showed early hints about Paradox.
In Trollz, the Trollz' pets, aside from Amethyst's dog, Wa-Wa, are only seen on the website. Some of the backstory involving Simon is only found there, too.
Young Justice had a tie-in comic, like most DC animated adaptations, but theirs was co-written by writers and producers from the series and went into details like why the Justice League abandoned their headquarters at Mt. Justice, how Artemis found the battle with A.M.A.Z.O., and just whySuperboy hates monkeys so much. It also provided an origin for Clayface, and at least one issue contained a scene that was scripted but ultimately cut from an actual episode of the show.
The producers have announced that the upcoming Young Justice: Legacy video game will shed some light on what happened during the five year Time Skip between seasons one and two, as well as introduce a new villain who would've appeared in season three, had it not been cancelled.
The Avengers: Earth's Mightiest Heroes has tie-in comics containing stories and bios that confirm such details as the Crimson Dynamo's real name, Ivan Vanko, and the identity of the "giant robot" Iron Man fought in his first episode, Ultimo. They also record the capturing of some criminals whose defeats did not receive inclusion in the show's 52 episodes.
The tie-in comic book for The Avengers: United They Stand attempted to make sense of some of the baffling elements of the cartoon, such as providing a decent explanation as to why the Avengers were wearing gaudy, Animesque battle armor. It also provided an origin for Ultron.
A lot of information in Rollbots on the various tribes and special information regarding characters can only be found on the Rollbots webpage on the Ytv website, which can now no longer be accessed. Even when it was available, many aspects of it could not be viewed from outside Canada.
Much information about the Teen Titans could be learned from the Teen Titans Go! comic book, such as Terra's origins and much of Starfire's family.
The Scooby-Doo gang's ages are never stated within the series, but they are within outside material. An official calendar from the original series pinned Velma at 14, Shaggy and Freddy at 17, and Daphne at 16. A more recent example – bios from a video game – based off the same series has Velma at 15. Sources also have their sizes as: Fred is 5′11″ and 185 pounds, Shaggy is 6 feet tall and 160 pounds, Velma is 4′9″ and 95 pounds, and Dapnhe is 5′7″ and 115 pounds.
Issue #21 of the Gold Key comic (story "Charmed Strife") posits that the gang have the following zodiac signs: Scooby, Aries; Fred, Pisces; Shaggy, Leo; Velma, Virgo; Daphne, Gemini.
Since Bravest Warriors is still in production at this point, a lot of the information about it comes from the production art.
On The Flintstones, when the Rubbles first find baby Bamm-Bamm in a tortoise-shell basket on their doorstep, they quickly learn that he has near-superhuman strength. No explanation was ever given for this strength on the show. Some 30 years after the series aired, a trading card finally gave an answer. Bamm-Bamm was raised by dinosaurs in the wild.
Many of the background ponies/minor characters are never named in-show, their names only being found on toy packaging or trading cards. This even extends to supporting characters' surnames/middle names; Trixie's surname is "Lulamoon" and Diamond Tiara's middle name is "Dazzle". The greatest example of this is probably the Changeling Queen: she's never referred to by name in the show proper, and her name, "Chrysalis", was relegated to the script until the comics came out and made it official.
The chapter books such as Twilight Sparkle and the Crystal Heart Spell contain a lot of worldbuilding that never showed up in the show. It also expounds a fair bit on Princess Cadance's backstory, something the show barely even mentions.
Likewise, in the show, Twilight basically states that Cadance was the best foalsitter ever, but we don't get to see how well they got along until the comics covered it.
Spongebob Squarepants: Do you want to know what anti–sea-rhinoceros undergarments look like? Just play the game based on "The Camping Episode".
There's a lot of info/backstory in Motorcity that wasn't revealed in the show, but revealed by the creator, crew and the series bible, such as Julie and Claire being Childhood Friends, Mike's Cool Big Sis Capri, and Julie's mother having died when Julie was little. Since some of the info in the bible (particularly regarding how Dutch and Texas interact with each other) contradicts what we saw in the series, some of this info might not be canon.
Poet Anderson: The Dream Walker, being a short film, has worldbuilding and character details that are left out due to time. Details such as a full blown conflict between Dream Walkers, the nature of their souls, and the main character being an orphan with a brother are only available in the prequel comics, interviews with Angels & Airwaves, and Tom DeLonge's Instagram.