The Arcane Trickster is a character archetype popularized, if not created, by Dungeons & Dragons. It owes its origins to the similar perception of both the wizard and the rogue as "intellectual" classes, courtesy of the latter's status as the iconic D&D Skill Monkey. Given this overlap, many players begin to wonder if it doesn't make sense for the rogue to try and pick up a few wizardy tricks to make their life easier, something reinforced by the way that, in both Basic D&D and Advanced Dungeons & Dragons, rogues gained a limited ability to use magic scrolls from 10th level onwards.
When AD&D's multiclassing rules allowed for the existence of the Mage/Thief, the Arcane Trickster was born; similar in nature to the gish, the Arcane Trickster as a concept is a character that combines stealth and thieving skills with magic. Unlike gish, who vary across the spectrum of magic to martial ratios, arcane tricksters are usually seen as more a full-fledged rogue who just dabbles in magic, to an extent... although this is somewhat dependent on the player and the edition.
The Arcane Trickster was first named when it made its debut as a Prestige Class in Dungeons & Dragons 3rd Edition. It subsequently reappeared as a Rogue Paragon Path in the splatbook "Martial Power 2" for Dungeons & Dragons 4th Edition. Finally, it appeared as a core subclass for rogues in the Player's Handbook of Dungeons & Dragons 5th Edition.
It's a 10-level prestige class that requires a Sneak Attack of at least 2d6 damage, the ability to cast Mage Hand and at least 1 3rd level spell, a Non-Lawful alignment, 7 ranks in each of the following skills: Decipher Script, Disable Device, Escape Artist, and 4 ranks in the skill Knowledge (Arcana).
An Arcane Trickster gains +1 level's worth of spells per day with each level it takes in this class. It gains +1d6 damage to its Sneak Attack at every 2nd level (2/4/6/8/10). At level 1, it gains the feature Ranged Legerdemain, which basically lets it use a mage hand to perform a Disable Device, Open Lock, or Sleight of Hand check from up to 30 feet away. It can do this 1/day, increasing to 2/day at 5th level and 3/day at 9th level. At 3rd level, it can make an Impromptu Sneak Attack 1/day, increasing to 2/day at 7th level.
A 4e Rogue with training in the Arcana skill can take the Arcane Trickster Paragon Path at 11th level. At 11th level it gains the features Arcane Burglary (you can use daggers as implements and substitute Arcana for Athletics) and Fading Action (when you spend an action point to make an attack, 1 target of that attack treats you as invisible until it passes a save) and the Encounter attack power Blinding Flare (Arcane power, attacks each creature in a close burst 1, hit creatures are blinded until the end of your next turn, and you can shift your speed). At 12th level, it gains the Daily utility power Cat Burglar's Veil (let's you conjure an illusory wall that makes you and your allies invisible to enemies so long as you are behind it). At 16th level, it gains the Dagger Spell feature (you can use your dagger as an implement to cast ranged arcane powers as melee weapon powers). Finally, at 20th level it gains the Daily attack power Confounding Teleport (can only be cast if you're holding a light blade, lets you teleport your speed to a creature and make an attack that does 4x your weapon damage + your Dex modifier on a hit, and half that on a miss, and afterwards you gain a teleport speed equal to your speed for the rest of the encounter; you have combat advantage against any creature you teleport adjacent to whilst using this power or using the teleportation granted as its after effect).
In 5e, it's one of the core Rogue subclasses. It turns your rogue into a third-caster who can learn Enchanter and Illusionist spells (wizard spell-list) of up to 4th level, with a mandatory Mage Hand cantrip, and grants you the following class features:
Mage Hand Legerdemain: You can turn your mage hand invisible and use it to perform sleight of hand, lockpicking and trap disarming. Your Cunning Action's bonus action can be used to control your mage hand. Gained at 3rd level.
Magical Ambush: If you cast a spell whilst hidden from the target, that target suffers Disadvantage on their saving throw. Gained at 9th level.
Versatile Trickster: As a bonus action, designate a creature within 5 feet; your mage hand starts messing with that creature, giving you advantage on attack rolls against it until the end of the turn. Gained at 13th level.
Spell Thief: Once per day, when targeted by a spell, you can force its caster to make a saving throw using its spellcasting ability score; if it fails, that spell has no effect on you, the original caster can't cast it for 8 hours, and if the spell is 1st to 4th level, then you can cast it using your own spell slots for the next 8 hours. Gained at 17th level.