From Norse mythology. Depending on whom you ask, he's an aesir, a jotun or both. A trickster god, he starts out as at least on the side of the aesir, but crosses the line when he engineers Baldr's death. He eventually leads the armies of the jotun during Ragnarok.
Loki serves as the main antagonist of Digital Devil Story: Megami Tensei. Akemi Nakajima summons Loki and has him get revenge on several students at Jusho High School. Loki seizes control of several students' bodies and murders the students that Nakajima wanted revenge on. Loki then demanded women be sacrificed to him, eventually allowing him to impregnate Ohara, one of Nakajima's teachers. While Nakajima was no longer watching, Loki practiced teleporting through computer screens and taking physical form, which he eventually uses to rape and mortally injure Yumiko Shirasagi, one of Nakajima's fellow students who happens to be the reincarnation of Izanami. Nakajima attacks Loki with Cerberus and then escapes to Asuka, where he plans to resurrect Yumiko. Loki, angry at being injured and fearful of Nakajima summoning other demon lords that could compete with him, follows Nakajima to the tomb with the hopes of killing him. Loki kills Yomotsu-Shikome and he causes Yumiko's body to rot and attacks Nakajima in an attempt to break his spirit. Izanami appears to fight Loki, but Loki captures her and prepares to rape her. Nakajima manages to summon Hi-no-Kagutsuchi as a sword and uses it to kill Loki.
Loki appears in Megami Tensei as a secondary antagonist. Lucifer resurrects Loki for him to aid him in the takeover of Earth, and will be confronted in the Valhalla Corridor. A wizard named Lor attempted to defeat him, but failed. His talisman, obtained from Gar after defeating Medusa, allows Akemi Nakajima to properly fight him due to sealing his magic. After defeating him, the talisman will shatter and Nakajima will obtain the Orb of Malice.
"Origin: Scandinavia. A beautiful blond giant with blue eyes. He fathered Fenrir, Hel, and Jormungandr with the giant Angrboda. His cleverness along with his beauty was the cause of a great deal of trouble. He later grew evil enough to have Odin's son Baldr killed. Without Baldr's light, the world began to progress steadily toward Ragnarok. There, he will lead the giants against the gods. He and Heimdall will kill each other."
—Shin Megami Tensei A-Mode DDS Dictionary
As an effect of his race, Loki is oddly categorized as "law" and "Light" aligned, an odd choice for the demon.
"A major deity of Norse mythology. He is known as a trickster and has had clashes with the other gods. He is a blood brother to Odin, but later was involved in the death of Odin's son, Balder, and was punished."
—Shin Megami Tensei III: Nocturne Compendium
Loki appears as an unrecruitable demon in Shin Megami Tensei III: Nocturne. He is also a barfly that frequents Nyx's lounge in Ginza.
He is approached in order to purchase a 1000-Yen Bill that the Collector Manikin in the Great Underpass of Ginza requests. However, taking advantage of human items' rarity in the Vortex World, he intends to charge the Demi-fiend 20,000,000 Macca for the bill.
Instead, taking advantage of Loki's room's back entrance, the Demi-fiend will slay his guardian, Troll, and ransack Loki's room and the several chests of valuables he keeps there. This eventually has some consequences if the player decides to pursue the Labyrinth of Amala quest, as Loki, to compensate for the loss of his valuables, will sell the key he was supposed to guard for more drinking money.
"The blond-haired, blue-eyed trickster god of Norse mythology. He was born of the race of giants, but was accepted as one of the Aesir by Odin. He is strikingly handsome, but has a cunning streak that makes him a frequent nuisance to others. His murder of Baldr, Odin's son and the god of light, robbed the world of light and signaled the beginning of Ragnarok, the end times. With the giantess Angrboda he sired three children: the giants wolf Fenrir, the world serpent Jormungandr, and the goddess of the underworld, Hel. At Ragnarok, Loki will lead the giants in their battle against the gods. There, he will fight the light god Heimdall, who had taken back the necklace Brísingamen after Loki stole it from the goddess Freya, and the two will slay each other."
—Shin Megami Tensei IMAGINE compendium
Loki can only be obtained through triple fusion, though he does not normally require a plug-in. He appears alongside Hel in the Second Home instance, but does not regularly appear as a boss. A variation with a different model was introduced through summon PG and later as a plug-in acquired through Shinjuku's Gomory in exchange for working for all five demons within the Pentalpha system.
Loki can be found in the Monochrome Forest within the Expanse. He can teach Flynn the Pulinpa, Mabufudyne, Taunt and Dekunda skills through his Demon Whisper.
Loki later appears in the Challenge Quest, Punishing the Trickster. He has retreated to the world of Infernal Tokyo and set up a Domain over Miyashita Park, loving how chaotic the world is. Seeing Flynn, he tells him to scram since he's in a good mood or he will tear him apart. Realizing the aesir, specifically Heimdall, sent Flynn angers him, leading to Loki cursing Heimdall for his persistence. He reveals he planned on relaxing for one hundred or more years, but since he has been tracked down, he'll show Heimdall what he's fully capable of and start a war.
Loki is shocked that he is defeated by a human and decides to forget his plans for a war. However, he enjoyed the fight against Flynn and didn't expect any surprises in the current world. He admits that surprises like a defeat at human hands is one reason why he could never quit being a god. Loki then laughs as he disappears and the domain vanishes.
Loki appears as a boss in the quest Pursue Hunter Hunters, where he tricked a group of Dormarths into believing he's a part of the Divine Powers. He reveals after killing the last one that it was a lie and he hated the Powers and wished to prevent them from getting the souls they desire. He decides to play with Nanashi in a battle and upon defeat hopes to one day fight him again in a joyful match when the world is set aflame, dying afterwards.
Loki can teach Nanashi the Pulinpa, Bufudyne, Agidyne, Ziodyne, Zandyne and Cold World skills through his Demon Whisper. He benefits from learning Fire, Ice, Electricity and Force skills.
Loki can be found in random encounters in late game dungeons, usually alone. He can use the Despair Night skill, which can drain levels of anyone without protection from death magic.
In the 3DS remake, a variant Loki is available from Nemechi, unlocked at the leg stage of the female growth path. He costs 300 D-Souls and uses his artwork from the first Devil Summoner installment. This version has an unusual gimmick: he starts out with very low loyalty, but has access to all four magical Loyalty skills. Once sufficiently bribed with gems, he becomes a very powerful elemental spellcaster, and is a convenient way to pass those skills to a dragon type Zeed. Due to this form's Sly personality, he can also be transformed into Joyous Bedding with Mystic Change.
"A major Norse deity, known as a trickster who clashes with the other gods. Blood brothers with Odin, but was later punished for his role in Balder's death."
—Persona 3 FES compendium
Loki, when paired up with Susano-o, allows the protagonist to use the Fusion SpellTrickster, which inflicts multiple mental ailments upon all enemies, alluding to their roles as tricksters among deities.
His combat traits remain unchanged throughout the subsequent iterations of Persona 3, however he gets a design update in Persona 3 Portable where he uses his Devil Summoner: Soul Hackers design instead.
Loki appears as the Ultimate Persona of the Fool Arcana in Persona 4. He is available to be summoned once the protagonist manages to gain access to the Normal Ending route, which maxes out the Fool Arcana Social Link. Loki also holds the distinction of being able to learn the most powerful Ice elemental skill throughout of the game, Niflheim.
In the True Ending special, Margaret summons Loki to attack Yu and help him regain his confidence in fighting against Izanami-no-Okami. Loki was defeated through the combined efforts of Makami of the Hanged Man Arcana, Abaddon of the Tower Arcana, Genbu of the Temperance Arcana, Mot of the Death Arcana and Beelzebub of the Devil Arcana.
Loki appears as the true Persona of Goro Akechi, manifesting during the second battle with him in Shido's Palace. His ability to induce states of berserk rage is revealed by Akechi to be the source of the strange epidemic of sudden assaults, a product of Akechi's experiments with the Metaverse. He is of the Justice Arcana.
Loki appears in the Dark Sun special, being summoned by Akechi to induce rage states into both Cu Chulainn and Cerberus. After they are defeated, he is summoned by Akechi and clashes with Joker's Persona Arsene. However, Loki is eventually overpowered and defeated by Arsene, resulting in Akechi's loss.
Loki disguises himself as an unnamed human being, known only as Gigolo by the in-game dialogue boxes. He is voiced by Yuji Ueda in the Japanese version of Overclocked and Vic Mignogna in the English version.
Gigolo's knowledge of the history of the demon world has helped the protagonist from time to time, most notably hinting towards the location of the Devil's Fuge so that the protagonist can defeat Beldr. Loki has no interest in the Throne of Bel, and does everything for his own amusement.
In the Desperate Escape ending, Loki appears in his human form and tells off the protagonist for attempting to escape the lockdown instead of fighting for the Throne of Bel. If the player failed to save Haru (and thus not need to fight Belial), he also calls the player a failure for "dropping out." After revealing his true form, he attacks the protagonist's party, and even creates duplicates of himself to confuse them.
In the Kingdom of Saints ending, Loki also appears while the protagonist faces off against Naoya. Upon seeing the two siblings fight, he finds it fit to join in for his own amusement, and reveals his true form.
In Overclocked, Loki will appear during the Naoya/Kaido's 8th day route, The King's Decision, in an optional encounter. Loki tricks Honda into luring the protagonist into an ambush in order to amuse himself, with the false promise of resurrecting Honda's son.
After defeating Loki in any of the above battles, he becomes available for the player to fuse.
"A major Norse deity, known as a trickster who clashes with the other gods. Though his parents are giants and enemies of the gods, he joined the pantheon as Odin's blood brother. Due to his countless evil deeds, such as the murder of Beldr, he was eventually punished with imprisonment in a cave. He is fated to be released on the day of Ragnarok, when the world will end and he will battle the gods."
—Devil Survivor 2 compendium
Loki has a small appearance as an enemy with dialogue. Polaris in his second phase summons him and in this incarnation, he talks like a rocker.
"An evil god of Norse mythology known as the trickster. Loki was born to the giants, archenemies of the gods. Thanks to a treaty that Odin, lord of the gods, made with the giants, Loki was permitted to join the ranks of the gods. He was said to look cunning, attracting many. At times he would even transform into a woman to have a male mate. Though he is the cause of much strife, he is known to be the hero in some situations which make him a peculiar and intriguing god."
He is the primary antagonist of the original Digital Devil Story: Megami Tensei novel, as well as the cult-classic anime OVA adaptation it spawned, and has the distinction of being the very first demon to appear in the entire Megami Tensei franchise.
Similar to Norse myth, Loki is portrayed as being rebellious and tricky; but at the same time, possesses a cruel and ruthless personality.
The sword wielded by Loki in Persona 5 is actually a pair of large pliers gripping a blade that is extremely hot, too hot to be touched.
While Loki could be considered Akechi's ultimate Persona, considering he follows the "mythological trickster" pattern of the party's ultimate Personas, he could also be considered his true Persona, considering Loki’s unique ability was used by Akechi 2 years prior to the events of the game to help Shido out of a scandal, as well as gain his status as the "charismatic detective." It wasn’t until he needed to act as a mole within the party that he began using Robin Hood.
This is further proven that in Persona Q2: New Cinema Labyrinth, Robin Hood does not awaken into Loki or anything whatsoever, indicating that Loki and Robin Hood are standalone.
However, data for Loki exists in the internal files of Persona Q2, including a compendium entry.
By hacking Persona 5, it is possible to equip Loki to either Akechi or the protagonist as a Persona. He is of the Justice Arcana, and his stats and base level are identical to Robin Hood's, but he lacks any skills so the player must add them themselves. The protagonist has no voice clips for summoning Loki, but Akechi does, indicating Akechi would have been able to use Loki normally. In addition, there are two unused duplicates of Loki in the game, but both of these are level 1.
Loki's model in Persona 5 has sharp red teeth, however these go unseen in game as Loki has no animations of its mouth opening. 
Regarding Loki's design in Persona 5:
"Loki casts a mysterious image. His design is a large body patterned in white and black, and we hoped that it would conjure up the image of not being able to see where his loyalty lies. The design is based on the "dazzle camouflage" of World War I warships. This camouflage was used to make it harder to figure out the shape or size of warships. The horns extending from the eyes were based on snail parasites, and emphasis was placed on making him appear sickening and with no apparent sense of direction. We thought it would be good if the sight of Loki made even Akechi question, “Who are you really?”"