You've seen it before. For lack of a better introduction, the main magic user in the game or story happens to be the youngest person in the group. Possibly because the story writer wants to have a character that young for comic effect and can't imagine said character being proficient in a sword or other conventional weapons, so they let them summon meteors with their mind. The lesson apparently being that magic is just that easy. Or They're just that good.
There's also the old standby that children have more magic potential than adults, presumably tied to their imagination/innocence/not-being-attached-to-the-mundane. As such, there's always a chance of Growing Up Sucks for a mage.
This trope extends towards the Little People sometimes, as well, considering many have a child-like appearance (or even mannerisms). Just don't say that to the dwarves.
See also Tykebomb, which is not necessarily a magic user, and Squishy Wizard, which a Child Mage is almost guaranteed to be, or Cute Bruiser, when the character actually does have physical prowess instead of magic, despite their size, and Goo Goo Godlike, for a character even younger (specifically, they are infants) and more precocious than the Child Mage.
If they have psychic abilities instead of magic it's Psychic Children.
Any Magical Girl series, especially ones where the characters are children below high school ages.
The main character of Mahou Sensei Negima! is incredibly proficient with magic for being only ten years old. He also is a partial subversion, as he becomes skilled with physical combat as well, but this is backed up with magic too.
The Big Bad is a subversion, as when he's first introduced he's apparently the youngest member of his team, and the most powerful magic user. Turns out that he's probably either an Artificial Human or Older Than He Looks.
Both of which have been confirmed.
Schierke from Berserk is the youngest member of Guts' new party. Her spells are incredibly powerful but needs time to cast (and she's in a trance whenever that happens), so Guts and the other fighters cover her while she gets ready.
Magical Girl Lyrical Nanoha, and several of her friends/rivals... she was nine when she started using magic, and yet, she's kicking ass in the interdimensional league.
Harry Potter, while having most of the cast fit the basis for the trope, have some of the children be more proficient than others. Harry himself, for example, has infiltrated the Ministry of Magic multiple times.
Then again, it's shown that most adults are far more proficient at magic than the kids, and Harry isn't particularly good or powerful at magic, at least outside of defense against Dark Arts. He's got otherthingson his side.
The Discworld series first used this trope with Esk in Equal Rites. A much more extreme example occurs in Sourcery with Coin, who is not only a wizard, but the most powerful one since the mage wars thousands of years ago because Coin is a sourcerer, which effectively gives him an infinite amount of magical energy, unlike ordinary wizards who have to work with the background magic field. Being around Coin (or on the same world as him) makes all the wizards a lot more powerful as they can use the magic he constantly leaks into the world.
The Archive in The Dresden Files, a young girl who serves as the embodiment of the collective knowledge of all mankind... including magic, as demonstrated when she takes down four fallen angels all by herself.
Harry's first spell was to propel himself really far in the long-jump at school. He fought off his first Eldritch Abomination and won his first Magical Duel by eighteen.
Elaine, DuMorne's other adopted child, was also this.
Molly's first spell was to try and break her friends' addiction to drugs. There's a reason that mind magic is usually punishable by death.
Zilpha Keatley Snyder's The Changeling keeps it charmingly ambiguous whether the three-year-old Josie is this or whether her sister and friend just convince themselves she is.
In The Inheritence Cycle Riders, Elves, and Dragons all live a really long time because of their magic. When Eragon is crushing on Arya, she explains that in her culture, she's the youngest of adults and he's barely out of diapers. Even ignoring the fact that it's his job to defeat Galbatorix and she's not helping with that and the fact that she's affectionate for him, she tells him to wait a few decades before she'd really consider it.
Ged is this early on in A Wizard Of Earthsea before he goes on to become Archmage. His aunt, a witch, notes that he has unusual magical power, and when he was eight or nine, he saved his entire village from the Kargs using a spell he essentially made up on the spot. Some time after, he goes to Roke, which, as it's the Hogwarts of Earthsea, is also full of child mages.
The Young Wizard Series by Diane Duane states this trope a good number of times in many books. Child wizards are naturally stronger because of their imagination and removal from a serious life. Older wizards balance this out with their knowledge and experience.
Petra, the younger sister of the protaganist of The Chrysalids by John Wyndham is the most powerful telepath of the group. It's the strength of her powers that bring the woman from Sealand to rescue the group.
In Septimus Heap, the titular character is an adolescent who has the makings of becoming a powerful wizard one day.
In the Sword of Truth, most of the wizards at the Palace of the Prophets are Really 700 Years Old because the palace was designed as a spell-form. Richard, and Zedd (despite him being one of the oldest characters in the series otherwise), are as children to them.
Zedd refrains from teaching Richard magic so that Richard will default to his instincts, which seem to be magically augmented by being a War Wizard and honed so that he was worthy of being named Seeker. It turns out to be justified; the conventional understanding of how magic works turns out to be limited, Richard has more powers than anyone in the last several thousand years and he accesses those powers in a different way. If anyone had explained their understanding of the limits of magic, he'd have doubted himself in critical moments and been unable to use his power.
In The Dubious Hills by Pamela Dean, all young children are "magicians", while older magic users are "wizards". Most magicians lose their powers around age ten, a few years before coming into their "knowledge", the one area or subject that they understand instinctively and completely; the exceptions are future wizards, whose knowledge is magic.
In the Wheel of Time, most Aes Sedai first use their power by their very early teens, and often have very selfish (and Nightmare Fuel) early Weaves that they cobbled together that have to be beaten out of them. One particularly resourceful Aes Sedai puts together how Compulsion works based on these fragments, some science, and being really lucky to have not blown herself up by accident.
Played around with in The Candy Shop War. While all the actual magicians are grown-ups (and nearly all are Really 700 Years Old, because nearly every magician knows how to freeze aging and it takes a while to learn magic), magic is much stronger when used on or by children, and it is all but impossible to reverse non-magical aging. For that reason, mages can grant tremendous power to children but not use it themselves. The plot of the first book hinges on this catch-22, as the villainess is trying to acquire a drink from the Fountain Of Youth. The combination of childish power and adult knowledge makes her a Physical God.
A grade-school aged male witch with psychokinetic powers appeared in the first season of Charmed. Once the Magic School was introduced, there would be several more. Wyatt would be an extreme example of this, having powers in utero.
Dungeons & Dragons both inverts this trope and plays it straight in 3rd edition - of the two major arcane spellcasting classes, wizards have the highest minimum starting age while sorcerers have the lowest. The idea being that Wizards gain their power through countless years of study, while Sorcerers have an innate gift, and so can cast magic at a much younger age.
Eberron has the Loli-Pope Jaela Daran, the head of the Church of the Silver Flame and a eleven-year old girl. She's a 3rd-level cleric on her own merits... but when communing with the Silver Flame, she's one of the most powerful spellcasters in the setting.
The Mystara setting used these in The Principalities of Glantri, a supplement describing a nation ruled by wizards. Because only wizards can inherit noble titles in Glantri, magic-user families teach their kids to cast spells at extremely young ages, so if their parents should die prematurely, the title will stay within the family.
And one B-list schemer described is one of these, a necromancer with a level beyond his age. It is noted that treating him as a child is a very bad idea.
While averting the trope with magicians, 3rd Edition Shadowrun had the Otaku- children capable of accessing the Matrix without any technological apparatus, who lost their abilities as they aged. 4th Edition changed this, however- the abilities stay with age and they're known as Technomancers.
The most powerful wizard in the Warhammer Fantasy world is the High Elf Archmage Teclis. His magical abilities were phenomenal and precocious even as a small child (as detailed in the novel Blood of Aenarion, where he is barely into his teens), and while he is all grown up in the modern age of the setting, he's still only a couple of hundred years old, which is practically nothing for an elf.
Rydia in Final Fantasy IV the first time she joins the party, though after being separated, she spontaneouslyages several years before she rejoins later on. From the same game, Palom and Porom are about five (and twins), and serve as your major magic source for about the second fifth of the game.
However, none of these three characters lose any magical prowess as they grow older (outside of the usual between-game level resetting), though none of them gain any physical prowess, either.
Rydia does lose her white magic after she rejoins the party and never gains acess to it again, however Rosa and Porom fill the white mage roll well enough. (a bonus dungeon in the GBA/PSP version where she returns to a child does restore it however)
Relm of Final Fantasy VI has the highest natural Magic stat in the game. At 10, she's the youngest playable human in that game.
Final Fantasy Crystal Chronicles: The party members are all depicted as youngsters no older than fourteen or so, based on interaction with their family members. This includes the magically-inclined Yukes, who despite their age, are around the same size as adults from other tribes...
Final Fantasy XIII: Hope, once he is given his l'Cie powers, is statistically oriented to become the party's Squishy Wizard. He is the only character in the game to learn every offensive spell.
Elize from Tales of Xillia and Tales of Xillia 2 is the youngest member of the group at age 12 (13 in the sequel). Her magical prowess stems from the fact that her living doll Teepo is actually a booster; a device that increases one's combat capability at the expense of their lifespan.
Ivan from Golden Sun is the youngest party member and he's the Black Mage. In the sequel, Sheba also fills in this role. In the third game, Himi is even younger at twelve.
Some Tales games play with this trope: Meredy from Tales of Eternia is 16, and looks about 12. However, on her home planet, not only is she considered an adult and and owns her own lavish house, she's one of the most accredited and famous engineers in the world. Beryl Benito from Tales of Hearts likewise looks prepubescent, but is actually 18, and simply revels in her youth.
MOMO from Xenosaga. She's the youngest party member, a robot girl who looks like a preteen (and may even be Younger Than She Looks) and the best healer, a user of ranged magic-like attacks, and a part-time Magical Girl. However, this is because she's an Artificial Human, not because she's a child.
The Fire Emblem series has many mages that are children.
Also Subverted in Radiant Dawn with two of the most important mages, Soren and Micaiah. they're both products of breeding between the two species, meaning they age slower. By the end of the second game, they're both in their twenties while still appearing to be in their mid-teens. Empress Sanaki and Tormod before he grew up, however, fit this trope.
The twins Yubello and Yumina, a mage and cleric respectively are, at 13-14 years old, the youngest party members you can have in Mystery Of The Emblem.
Ricken from Awakening certainly evokes the trope. Although his age is never given, he is the youngest member of Chrom's Shepherds and is of marriageable age. This can be somewhat amusing as it is possible that his child from the future will look older than him.
Leon of Star Ocean: The Second Story is the youngest potential recruit, being only twelve, and is a prodigy who excels at magic. He's also a bit of a jerk, though he softens up after having a Heroic BSOD after his parents and everyone save Claude appear to be wiped out in a shipwreck. This is also the point where he becomes permanently recruit-able, though he's only accessible if you're playing as Claude.
Toyed with in the mage origin of Dragon Age: Origins. The campaign starts with the player character finishing his/her Harrowing, marking him/her as an adult and fully-fledged mage, but is noted to be young to be taking the Harrowing, and is almost certainly the youngest member of the party aside from perhaps Alistair, who himself seems to be in his early twenties.
Arl Eamon's son Connor is a straight example. As a very young mage, he can barely cast simple spells, but is just as capable of not so accidentally tearing the Veil to the Fade (the realm of demons, spirits, and dreams) as any other mage.
Skulls in Disgaea games are shown to be young boys. Their female counterparts who also appear in other non-Disgaea Nippon Ichi titles look no older than grade school aged as well. (Interestingly enough, the Cleric class is always depicted as older, and the female Clerics in particular much better stacked.)
Playable mages of either gender in Dungeon Fighter Online look like children, and females usually have the Genki Girl personality to match. As all PC mages are nonhuman immigrants (and the males may be undead), this may be only skin deep, and roleplayers often play them as adult. The mage character in the Anime of the Game is of similarly uncertain maturity.
Alvin from The Witcher has an inherent gift that makes him the most powerful mage known. Unfortunately it's so powerful it's a hazard to himself and others, and he needs help keeping it under control. In the books, Ciri had a similar role, but she managed to seal it and swear off magic when she grew up.
RuneScape has Kennith. He's first met as a child in a quest. The next quest in the series, he's a young adult, and together with two other characters and the player he forms a party of four. The player's age is discussable, but Kennith most probably is the youngest member of said party. And he can cast even the high-level spells without any effort or even material components usually needed for casting.
Dragon Quest V has two - Bianca and Madchen. While Bianca is not that strong until she grows up, Madchen, however, has no excuse for being able to cast powerful ice spell and transform into a dragon at the age of 8.
Played with in Simon the Sorcerer games, where Simon is a regular British kid from our world who ends up in the magical realm. However, he doesn't have any magical abilities on his own.
Ernie Eaglebeek in The Spellcasting Series may possibly be considered this, except he's in his late teens and has plenty of sex, something not normally associated with a child mage (especially a nerd).
Ram and Rom from Neptunia. Gust fits the bill as well, but only in appearance.
Pearl Fey, if spirit channelers count. She's only nine and is already at least as powerful as her eighteen-year-old cousin who is of the more powerful 'main family' bloodline rather than Pearl's 'branch family'. If Maya was out of the way she'd probably be first in line to be the Kurain Master. Which is why Pearl's mother keeps trying to kill Maya.
In Haru-Sari magic can only really be used by "elves", people born with a mutation, either naturally or more commonly artificially induced. The downside of that power? Elves physically stay prepubescent children forever, at least until their thirties — when their magic has poisoned their bodies enough to kill them. And that does not even count in the common psychological disorders and bad social standing.
The children of Hathor the Cow Goddess are unusually smart. But this is just to illustrate how well she thinks unschooling works and to make her adult Straw Loser opponents look even dumber by comparison.
The students in Wizard School are already experienced mages — as opposed to Graham, the adult, who is clueless. Alas, everyone is subject to Good Is Dumb.
Explored on a similar vein in an episode of Justice League Unlimited, with a kiddified John Stewart unable to settle on what he wants until Wonder Woman yells at him.
There's also an episode where Terry has to rescue a psychic little girl from an evil organization of telepaths who want to initiate her via kidnapping. While he saves her, he also gets her help since she's a pretty powerful kid.
In Gargoyles, in the backstory of "Long Way to Morning," the Magus was both very young and a good enough mage to cure the prince. (Granted, he did need the Grimorum to do it...)