She may be small, but don't underestimate her. This spunky little thing is packing enough heat to take down a Swiss Bank, and that's just with the spells she learned in kindergarten.
The Black Magician Girl is a character type frequently found in fantasy that is primarily an offensive magic user. In personality, she is likely to be immature and outgoing, frequently a Genki Girl and Tomboy. This is carried over to her design, which is far less likely to be as elaborate or impractical as the Lady of Black Magic. She's also far less likely to be Ms. Fanservice than the Lady.
Physically, the Black Magician Girl is weaker than the fighters, often being a Squishy Wizard or Glass Cannon and being unable to attack for much, if any, damage. Her weapon of choice is usually a rod or staff, although ranged weapons are not uncommon, nor are improbable ones such as books. Her primary role in combat is to stand back and pelt the enemy with Fire, Ice, Lightning, not to get down inthe mud, the blood and the beerwith the melee types.
She will often be younger than The Hero, both physically and emotionally. Due to this, she will almost never be a love interest for him.
Black Magician Girls are, as the name would suggest, usually female, but male examples do exist. Black Magician Boys will be as immature as their female counterparts and may be physical cowards.
Subtrope of Black Mage. Compare to Lady of Black Magic, the other character archetype for Black Mages found in fantasy. In personality and skillset, this trope is the inverse of the White Magician Girl, for which this character type is often a Foil. See also Black Magic, which is magic that is evil, and White Mage.
NOT to be confused with Dark Magical Girl.
Schierke of Berserk could qualify, being a powerful witch and signifigantly younger than most other characters (except Isidro). However, her personality is almost a complete aversion, aside from tending towards being Skilled, but Naive.
Aika from Skies Of Arcadia defines this personality-wise, despite arguably not being the best spellcaster.
Final Fantasy IV has Rydia, who starts out a child but undergoes a Plot-Relevant Age-Up. To an extent, Rydia seems generally boisterous, but this is tempered by her past as the sole survivor of her Doomed Hometown. She also isn't in love with any of the other heroes, but does serve as the unrequited love interest of Edge and comes to see Cecil as a father figure.
Also Palom is a Black Magician Boy.
Vivi is a more contemplative Black Magician Boy than most of his female counterparts, but he packs more magic than any other mage into a body half their size!
Female mages in Fire Emblem tend to either be Badass Bookworms or start as White Magician Girls before becoming Magic Knights through promotion, but there are still a few true Black Magician Girls (generally thunder or fire mages), including Tiltyu from the 4th game, Nino from the 7th, and Lute from the 8th. Empress Sanaki from the tenth (appears in the ninth game as well, but only in cut scenes). True to the above statement, her specialty is fire magic, but she's also skilled in thunder, wind, and light... and the only fully-promoted magic-user who can't use staves. And yes, she has the personality for it, much to Sigrun's consternation.
Touhou's Marisa Kirisame fits this trope to a T. Loud, brash, and a pretty capable thief who uses her lifespan as an excuse for never returning anything she steals. Yet she's still one of the heroes of the setting, relatively speaking.
Aht of Radiant Historia fits the personality type of this trope so well you'd think the writers read this trope first. Her primary combat role is powerful trap and Area of Effect spells. She also comes with decent healing magic, but she's not as good in this role as White Mage Marco.