Since a lot of people were asking for a summary, this is a filler-free summary by arc of season 1, with no manga spoilers past DGM Hallow episode 1.

If you have the time, I highly recommend watching the first season. DGM is my favorite anime/manga franchise of all time and my summary will not do it justice.

http://www.animefillerlist.com/shows/d-gray-man
http://dgrayman.wikia.com/wiki/Story_Arcs
Season 1
Introduction

Detailed Summary: Allen Walker is a brown-haired boy with a dysfunctional red arm. He was adopted by Mana Walker, a traveling clown, after they met at a circus in England. After a hard life in the circus as an abused errand boy, Allen loved Mana and his new life together.  When Mana died, a figure called the Millennium Earl approached Allen and offered a contract to bring back Mana from the dead. Allen happily accepted. Little did Allen know that the Millennium Earl preyed on people’s grief to create akuma, in which the souls of the deceased were called back and trapped in the skeleton of the living greiving human. After murdering and taking over the human skeleton, the akuma were forced to obey the Earl. Mana’s soul returned, but cursed Allen for calling him back and attacked him to take over his body, Allen’s red arm instantly transformed into a claw and destroyed Mana’s soul. Mana’s curse left a pentacle scar on Allen and his hair changed into stark white from his shock of destroying Mana’s soul.

Afterwards, Cross Marian came and took in Allen as his disciple. He revealed that Allen’s arm was “Innocence”, a weapon considered the Crystal of God and it was the only way of defeating the akuma, as well as exorcising the souls trapped within them. Mana’s curse caused Allen to develop the ability to see and hear the trapped suffering souls crying within Akuma, which helps him identify and exorcise Akuma, which otherwise are disguised as normal humans. Allen grew up as a polite young man who gave off the impression of being a “proper English gentleman” and has a habit of using formal language at all times. Unfortunately Cross had a bad gambling, womanizing, and drinking habit. He would leave all his debts in Allen’s name resulting in Allen having to earn money in shady ways during his time with Cross. Cross trains Allen harshly, giving him humorously impossible tasks that leave Allen in fear of Cross and traumatized. One day, Cross told Allen that he was ready to become a true exorcist and to go seek The Order. Cross refused to go back with him to headquarters, and instead knocks out Allen with a hammer and disappeared, leaving behind his yellow golem Timcampy with Allen to lead the way.

After traveling to the Order alone with Tim and misadventures along the way, Allen arrives and first meets Kanda Yu, Lenalee, and Komui Lee. Due to a misunderstanding regarding Allen’s arm, Kanda attacks Allen believing him to be an akuma intruder. They have bad chemistry from the first meeting and continue to bicker throughout the series. Allen gets along with everyone in his new home, befriends the chef Jerry, and members of the Science department such as Johnny, the guy with the glasses. He vows to live by Mana’s saying “Keep on walking” and swears to save the Akuma souls by exorcising them.
Short Summary: Allen Walker’s adoptive father Mana Walker died. The Millennium Earl offers to bring him back to life as an Akuma. Allen destroys Mana’s soul with his Innocence arm. Cross Marian takes him as a disciple and sends him to the Order to become an official exorcist.

Ghost of Mater arc+Kanda

Detailed Summary: On Allen’s first mission, he gets paired up with Kanda. The Order investigates any strange reports because it might be related to Akuma and Innocence. Innocence has surreal effects on objects and causes unnatural phenomenon. The Order is fighting a war to destroy all the Akuma, find all 109 shards of Innocence,  and recruit more Innocence-wielders known as Exorcists. The Akuma in turn want to destroy all Innocence and the Order. In addition to the Exorcists, volunteers called Finders and researchers at the Science department assist the Order. At the abandoned Italian town known to be haunted by the Ghost of Mater, Kanda and Allen are sent to meet with the Finders and investigate reports of an Innocence animating a doll. Kanda’s pragmatism and willingness in principle to sacrifice the Finders and other Exorcists to complete the mission conflicts with Allen’s emotional approach as he tries to save the souls within the Akuma through exorcism as well as all the human bystanders. They find dead Finders and many Akuma attacking Guzol, an ugly old man implied to be the doll and Lala, a young girl protecting him. Guzol and Lala run away. The attacking Akuma are revealed to evolve into a stronger Level-2 form, which nearly kills Kanda with a trick attack. Nearly defeated, Kanda is revealed to be motivated by having someone he must desperately find before he dies. Allen is also heavily injured. They fall through the crumbling town and find Lala and Guzol. Through a plot twist, it turns out the young girl is the ageless doll. Guzol was left behind in the abandoned ruined town as a young boy eighty years ago because of his deformities. The singing ragged doll Lala was left behind by its previous inhabitants. She became known as the Ghost of Mater and killed all the children dropped off there out of rage at their horror on meeting her and being denied a reason for existing as a doll made by humans. Lala offered to sing the boy a song and planned to murder her him like the others if he denied her. Instead, he cried with joy at finding someone willing to sing with him and he took care of Lala, living together in Mater ever since. Lala wants to stay with Guzol who terminally sick until he passes away, but Kanda demands to take the Innocence keeping the doll alive and return it back to headquarters as soon as possible because of the danger and saying that in the war, some things must be sacrificed. Allen defends Lala’s request, saying that he will stay even if it requires Allen becoming the sacrifice instead. His sacrificial mindset angers Kanda. They are attacked by an evolved Level-2 Akuma. Allen and Kanda are severely injured. Allen’s arm upgrades into a new form and together they barely defeat the Level-2 Akuma. During the attack, the Innocence is taken from Lala and she is reverted to a broken doll. After the Innocence is returned, the doll no longer has her 500 years of memories, but offers the dying Guzol a song as she first did when they met eighty years ago. Kanda recovers quickly from his severe injuries. Guzol dies, but Lala continues to sing a lullaby for 3 days until she breaks down. She thanks Allen for allowing her to keep her promise. Kanda gives Allen the task of bringing Innocence back to headquarters while he moves on to another mission.

Short Summary: Kanda and Allen are sent to Mater on a mission. Kanda is disgusted by Allen’s sacrificial mindset and is revealed to have someone he must absolutely find. Together they recover an Innocence shard from Mater. Allen wants to become a" destroyer that saves". Allen’s claw gets an upgrade into a gun form Cross Paling. Exorcists are revealed to have a synchonization rate that reflects their bond with their Innocence. Akuma is revealed to evolve, level up, and become stronger.

Rewinding City arc+ Lenalee, Miranda, Road
Detailed Summary:
Allen is sent on his second mission, this time with Lenalee who is the younger sister of the branch chief Komui Lee. They are sent to investigate a town with strange rumors. On arriving, they quickly find that they can not escape the town and the entire town resets at the end of the day, without any of the citizens noticing. Unlike before with Kanda, Allen and Lenalee make a great team and investigate the town. They meet and identify Miranda Lotto, an unconfident and nervous woman, as the source of the disturbance. Miranda is self-described as having been useless and awkward during her entire life, never making any friends, getting fired from every job, and even being mocked by the village children each morning in the rewinding town. One day, she noticed an old grandfather clock outside a shop, which the shop keeper explained was broken, useless, and unfixable by anyone who tried. Feeling sorry for it, Miranda tries turning the key to the clock and suddenly the clock starts ticking. Overwhelmed by something accepting her for the first time in her life, Miranda bought the clock. Allen and Lenalee realize that as exorcists, they are unable to physically touch the grandfather clock and instead pass through it.  Allen and Lenalee suspect that the clock must be Innocence and that Miranda’s feelings of uselessness are causing time to rewind. After she got fired from her 100th job, she wished that night for “tomorrow to never come. Each night, all of the previous day’s events are sucked back into the clock. The next morning arrives, Miranda wakes up not remember how she fell asleep and in despair at repeating another day.  They try urging Miranda to wish time to return to normal, which fails to solve the problem. They attempt to help her feelings of uselessness by getting a new job. Through odd jobs, including selling tickets on the street, Allen and Lenalee become closer. Allen is revealed to have circus training and learned clown tricks from Mana. Lenalee shares her past. Her parents were killed by Akuma and she was taken away from her brother after she was discovered to be compatible with Innocence. Imprisioned by the Order and increasingly restrained after each escape attempt, Lenalee grew to hate the Order, until her older brother Komui, who joined to Order to find her, was finally reunited when he was transferred to the European Branch. Meanwhile Miranda keeps failing at her jobs, from clumsiness and ineptitude, and breaks down after being called useless yet again, crying out why her clock turned out to be Innocence. Overheard by a strange girl named Road, Miranda get abducted, while Allen and Lenalee are ambushed by three Akuma and knocked unconscious. Road writes a message in Miranda’s blood, F You Exorcists. Allen wakes up with his arms nailed to the wall in a horrific and surreal dimension. Lenalee is unconscious and strangely garbed in gothic clothes. Miranda’s hands are nailed to her clock. Allen notices that Road is human and his cursed eye fails to see her as an Akuma. Shocked, Allen asks why a human could be siding with the Akuma. Road taunts Allen and retorts that weapons are used by humans to kill other humans. She claims the Millennium Earl as her brother, that the Exorcists serve a false God, and instead the Noah Clan are the true apostles. Rero, the Millennium Earl’s umbrella, chastises Road for telling outsiders and going outside of his instructions. Road plays on Allen’s hesitation to another human being. She embraces him to show she is human, then destroys her own face with his claw. As he recovers from his shock, she crushes his eye with a nail. Road then attacks Miranda expecting Allen to defend her. Allen does, but falls from his injuries. Miranda gains the resolve to fight and activates her Innocence, which turns out to rewind all injuries and halt time on further injuries, by storing them in her clock. Lenalee is also healed and together they battle Road. Road reveals she has a special interest in Allen after finding out that he wants to save the Akuma. She orders an Akuma underling to self-destruct, mentioning to Allen that the soul can’t be save unless it is exorcised. Allen rushes to exorcise the Akuma, but he is stopped by Lenalee moments before it explodes. Witnessing the soul begging for his help and disappearing, Allen snaps at Lenalee, asking why she stopped him. She slaps him saying he’s her friend. Allen and Lenalee destroy the remaining Akuma and Road retreats saying that she’s had her fill of fun. She opens a dimensional door, but Allen places his Cross Paling on Road’s head and aims to fire. She teases him saying that she knows he’s too nice to shoot, that his arm is a weapon, and that him shedding tears over an Akuma’s destruction is a “no-no”. She escapes, and after leaving Road’s dream dimension, Miranda’s powers are shown to only temporally story injuries and is physically demanding on Miranda to maintain her innocence. Allen and Lenalee persuade her to let go, and their tremendous injuries return. The Rewinding Effect on the town stops and Miranda leaves to seek help for Allen and Lenalee’s injuries.

Short Summary: Allen and Lenalee leave on a mission. They discover a new exorcist Miranda and her ability to temporarily heal injuries by holding time.  Lenalee shares her past. Her parents were killed by akuma and she was taken in by the Order due compatibility with Innocence. Her brother, the current European chief Komui Lee, joined the Order for her sake. Road Kamelot and the Noah Clan is introduced as the mastermind behind the Akuma attacks. The Noah Clan are human. Road as a sadistic personality.  When Allen makes a suicidal attempt to save an akuma soul, Allen snaps at Lenalee for stopping him, and in turn she slaps him. Road has her fill of fun and leaves. Allen and Lenalee are severely injured by the end of the arc, which Lenalee becoming comatose and Allen having a crushed eye.

General Yeegar Falls arc
Allen wakes up, which his injuries patched up and his eye completely crushed. He meets an old asian man called “Bookman” and his apprentice Lavi.
Vampire of the Old Castle Arc+Lavi, Krory
Order in Crisis Arc
Suman Dark’s Fallen Ark+ Lenalee
Asian Branch arc+Bak, Foh
Noah Arc