The Wild Mage is a form of arcanist originating from Dungeons & Dragons, having first appeared as one of two new wizard kits in the Advanced Dungeons & Dragons 2nd edition sourcebook "Tome of Magic". As their name suggests, these arcanists specialize in the use of Wild Magic, making them much more chaotic and unpredictable than normal arcanists.
Introduced and named in this edition, the Wild Mage is a specialist wizard of a different stripe indeed. These specialists receive the usual bonus spell splot for each level (which can only be used to memorized wild magic spells), but have no opposition schools. They have a +10% bonus to learning wild magic spells, a -5% penalty to learning other spells, and when inventing a spell, treats its spell level as being one level lower to determine how difficult it is to invent. They also have the unique ability to influence wild magic-based magical items; this gives them a 50% chance to control the outcome when using certain random chance-based magical items, even artifact-tier ones like the Well of Many Worlds and the Deck of Many Things.
What's the catch? Two things. Firstly, the actual efficiency of their spells is highly variable; when you cast a spell, you roll a D20 and compare your wild mage's level and the result of the die roll to a grid to determine the potency of your spell. Secondly, certain results on that grid will trigger a Wild Surge, a D100 check that, in addition to the spell going off, causes a random effect to occur. This can be silly, dangerous, helpful or annoying, with possible occurences including permanently destroying a magical item held by the caster, summoning a Rust Monster, creating a Gate to a random spot on the Outer Planes, casting a Heal spell on the caster and everyone within 10 feet, causing you to fall in love with your target , and causing the spell to be cast at 200% effectiveness.
This, naturally, makes the AD&D Wild Mage a very unpredictable and untrustworthy class; whilst many fans found it notable, not so many found it playable. Wanting to play a wild mage in AD&D isn't quite the same level of That Guyness as wanting to play a kender or a Xaositect, but it will earn you some distrust. or lulz.
Th Wild Mage was a Prestige class in third edition, presented in the Complete Arcane splatbook. A somewhat safer version of the second edition; subject to random modifiers on their abilities, but with less chance of things going horribly wrong.
Wild Mages have a random caster level, equal to their original arcane casting level minus 3 but plus D6, giving them a number anywhere between minus two or plus three, but otherwise averaging about equal to what they would have had. From second level they can start randomly deflecting incoming ranged attacks or spells that solely target the Wild Mage, they can do this as an immediate action without preparation and it lasts until their next turn, but can only use it a few times per day. The deflected attack or spell instead affects an random target within 20 feet and can even strike your allies, but it effectively renders you utterly immune to ranged attacks and spells of any level whatsoever until your next turn begins.
From third level you gain better odds when using magical items that produce magical effects (such as Rods of Wonder) and can roll twice and choose your preferred result.
From ninth level, you can start pulling the random crap that Wild Mages used to do, being able to convert your stored spells into random effects drawn from the Rod of Wonder table. This can be an ideal way to draw usefulness from low level spell slots when your character has long outgrown them, as the ability to throw a lightning bolt or turn yourself invisible instead of 1st level spells is awesome. Your third level ability also applies when you use this, so you can roll twice and choose your preferred effect whenever you commit to casting a randomised spell. Although you still run the risk of turning yourself a funny colour, or fooling yourself into believing the spell has gone off and stand there going "pew pew", you'd need to roll two bad results in order to really suffer from this. But at least it's not as bad as nuking yourself accidentally.
The Prestige Class also makes you immune to Confusion and Insanity spells as well as making you difficult to detect magically.
The final feature of the class gives you the ability to inflict wild magic on an enemy spellcaster, forcing them to use randomly generated effects whenever they use their own spells, 50% of the time. They get no save against this (though Spell Resistance applies), so you can really screw up a high level opponent's ability to destroy you when he can't reliably cast his own spells.
In 4th edition, the Wild Mage was resurrected in the Player's Handbook 2 as one of the two magical origins for the sorcerer. These sorcerers were naturally gifted in the art of wielding arcane energy, but lacked the finesse of other, more "refined" bloodlines, such as sorcerers wielding Dragon Magic. Whilst heavily toned down compared to their AD&D depiction, these wild mages still had a strong "random" theme to their abilities. Chaos Burst meant that their first attack roll of the round would grant them either +1 AC or a free saving throw, depending on if the roll was odds or evens. Unfettered Power meant that the Wild Mage added a slide & knock prone secondary effect to any attack power they rolled a natural 20 for, but would also Push all creaturies within 5 squares of them back 1 square on the roll of a natural 1. Finally, Wild Soul gave them resistance to a randomized elemental damage type (determined by a D10 roll) and the ability to ignore that damage resistance type with their spells, with the type of elemental damage being rerolled each time the Wild Mage completed an extended rest.
Sorcerer spells themed around Wild Magic tended to have randomized "odds or evens" type effects, and were usually more potent in the hands of Wild Mages.
An article in Dragon Magazine #385 played up the associations of wild magic with chance by creating a new "substyle" for Wild Mages, called Luckbenders; new feats, powers and a Paragon Path all added up to the ultimate gambling mage, able to play chance to its own advantages.
The 4e version of the Wild Mage was so successful that it was repeated in the D&D 5e Player's Handbook. This version was a bit of a mash-up of the AD&D and 4e versions, most notably by bringing back the Wild Surge and making it integral to some of the subclass's features. This version's Wild Surge table is much shorter and far less destructive than the AD&D version.
The Barbarian also got a Wild Mage subclass, ala the Bloodrager, in the form of the Wild Soul in the August 2019 issue of Unearthed Arcana. It can Detect Magic Constitution modifier times per day and gains one of eight random wild magic auras when it rages, from 6th level can take force damage to restore spell slots or grant temporary HP to allies, from 10th level can slap enemies that hit them with a Saving Throw effect whilst they rage for Force damage, and from 14th level can use their bonus action whilst raging to reroll which wild magic aura they get.