Wild Mage
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.
Wild Magic?
To better explain wild magic, one has to understand that wizards and sorcerers in D&D by tapping into the raw energies of magic, shaping it tightly to their will, and unleashing it in specific patterns of rigidly controlled force. The wild magic user, in comparison, just grabs it, roughly shapes it into the desired format, and then throws it out at his foes like a hot potato. This makes for both a strength and a weakness; wild magic is often more potent than the carefully controlled spells used by most arcanists, but it also has a tendency to behave... erratically. This can make wild magic users almost as dangerous to their allies and themselves as they are to their enemies.
AD&D
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, 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.
4e
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 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.
5e
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.