Aerith and Bob: In a way: most angels and demons have a proper angelic/demonic name and a common Japanese first name (Azazel Atsushi, Beelzebub Yuuichi, Undine Megumi, Salamander Kimitake, Moloch Yoshinobu, Zeruel Takeru, etc.). The angelic/demonic names function as surnames.
Indeed, while Sakuma does get along with Azazel, Beelzebub and most other demons, they don't particularly like her or care about her well-being, and they're sometimes seen plotting her and Akutabe's demise.
Animation Bump: Episodes 5 and 6 feature looser-than-usual animation direction by Chiaki Nakajima, who in the past has worked on The Tatami Galaxy and the token realistic Art Shift episode of Panty & Stocking with Garterbelt. As a result both episodes are some of the best looking in the show, with 6 having some of the most expressive and lively character movement in the TV series, not to mention some interesting scene composition and use of color.
AngelsAre Useless: The angels are just watchers unless they're fighting demons.
Archangel Lucifer: Lucifer is apparently not the king of the demons, but he is the highest ranking and probably most powerful demon. And he loves reminding others about this.
Ass Shove: Azazel attacks Sakuma with his pitchfork in this manner in the first episode of the second season.
Baleful Polymorph: If you abuse or damage a grimoire, you get turned into an animal. Okada gets turned into a chubby lizard while Seiya becomes a praying mantis. These transformations mean the transformed quickly lose their memories as a human too.
Sakuma is nearly turned into a cow by Moloch (in the first OVA) when she fails to provide him an adequate sacrifice.
Later she actually turns into a panda when Behemoth decides he doesn't want her sacrifice after all. She only turns back when Behemoth accidentally destroys the sacrifice, a PSP.
Berserk Button: Do not question Beelzebub's... peculiar eating habits.
Bratty Half-Pint: Koutarou, with the exception being that he dug himself out of his own mess when Sakuma and Akutabe didn't feel like helping him out.
Breaking the Fourth Wall: A number of times, most obviously when Lucifer specifies his sacrifice (marijuana leaves) and Azazel worries about the manga influencing the readers.
The Behemoth story, big time. A manga editor at Evening (the magazine Azazel-san is published in) wants to murder Kubo Yasuhisa and the entire plot revolves around Evening. Sakuma even likens a manga with half-assed artwork to Hunter × Hunter.
Brought Down to Normal: Inverted; Akutabe allows Beelzebub to take his normal form in the real world by releasing the barrier that makes them into cuddly plush toys. It lasts for two episodes, and then for the rest of it outside the Demon World he's a toy again.
Butt Monkey: Sakuma and Azazel, the latter even more so.
Call Back: In the first episode of season 2, we see a praying mantis and lizard in seperate containers, two previous Baleful Polymorths from season 1.
Council of Angels: Uriel, Raphael, Gabriel and Michael, four (Arch?)angels who seem to be by god's side at all times in the form of four various types of women (The Stoic, Gainaxing, The Pollyanna and Gonk). They are also in charge of making angels fall, cutting off the angel's wings and destroying their halo, before God banishes them to the human world.
Cosplay: Sakuma's forced into this so that she can get some notes for a test. She later joins for real once she finds out there is a profit to be made.
Crapsack School: Happens to Golgotha Middle School thanks to Undine.
The whole setting of the series can probably qualify as one.
Cute and Psycho: Beelzebub usually comes off as a pleasant (in his own way), well-mannered person... until he slips into the darker side of his persona, which is foul-mouthed, vulgar and violent.
Cute Little Fangs: Azazel's girlfriend, Kiyoko in her humanoid form. Azazel himself has them as well but they're not very cute.
Dark and Troubled Past: Many members of the otaku circle Sakuma joins have suffered severe traumas due to bullying and other abuse in the past.
Deadly Gas: In his monstrous form Beelzebub's breath is comparable to really, really noxious gases. It shreds/dissolves a normal demon's skin.
Demon Lords and Archdevils: There is a hierarchy in Hell, it seems, with different demonic families having one chosen representative with a special power. This representative is the one who gets summoned by a demon summoner. If one dies due to an angel or other ways, the family chooses a new representative to replace the previous one. Some families seem to be more royal than others, like Beelzebub's, but Lucifer just seems to be the strongest Demon there, rather than a leader. Being royalty seems just a title which doesn't mean anything practical.
Salamander's spit is awfully thick. And white. And sticky.
Sariel's hairstyle makes his head look like a penis.
Strawberry Warrior licking the condensed milk off the strawberry in the mock opening...
Deranged Animation: The series is mostly realistic, as per Production I. G.'s usual art style. Oftentimes the animation goes nuts though, almost to Ren and Stimpy levels. Off Model is used to indicate certain changes in someone's personal (Zeruel's animated face seems to literally collapse into a mirage of colours when he finds out he doesn't have a grimoire, and Akutabe is animated (And drawn with a weird expression in the manga) to just a choppy human shaped thing when he tells Undine he loves her.
Executive Meddling: With the manga being what it is, the anime has been on the receiving end a lot during production to be fit to air on TV. In interviews various staff members asked Kubo Yasuhisa to write stories that can be adapted easier for TV.
God: Has a sense of humour and has told angels on earth to find grimoires. Appears as just a shadow outline of a person on a throne, and wants to find someone who is more funny than him.
Gratuitous English: Mostly the American "sumo wrestler" in volume 4, but Salamander also has a couple of instances. And Lucifer's tattoo says "God /bless/ fuck your ass".
Hey, It's That Voice!: The anime has a pretty impressive cast, for its genre and category. In the voice commentary the director said he'd been surprised that some of the more high-profile seiyuu (such as Rie Kugimiya or Keiji Fujiwara) accepted his offer. We also learn that at least one famous seiyuu refused to even audition when he learned what kind of anime it was.
Impossible Shadow Puppets: God seems to spend his time playing about with this. He is literally just a shadow, and he makes reasonable shapes (A penis, a barking dog, etc) but can apparently emulate defecating and turn his face into a pair of breasts which jiggle.
Inconvenient Summons: The demons are often in the middle of something when they get summoned.
Played with when Sakuma summons Beelzebub while he and Lucifer are duking it out at full strength, saving the world of demons from serious damage.
The third time Azazel is summoned, he's having sex. The first time Beelzebub is summoned, he's on the john. Which actually makes for a good case of foreshadowing.
Invisible to Normals: The demons by default. They can make themselves visible at will.
It's All About Me: Lucifer. Oh Lucifer. But then, he represents Pride.
The sheer size of his ego makes him one of the few demons who are not intimidated by Akutabe.
Karmic Transformation: Happens to anyone who betrays the contract with their contracted demon without the protection of a grimoire. Like Okada and Himoi.
No Celebrities Were Harmed: Salamander's entire character is a parody of Yukio Mishima. (Although obviously he doesn't look like Mishima, being a small, red lizard, his fundoshi-and-katana outfit is a Shout Out to Mishima. His full name is Salamander Kimitake. Kimitake was Mishima's real first name.)
Otaku: Sakuma grudgingly cosplays for some in her university to get her hands on some notes.
Pandaing To The Audience: Behemoth looks similar to a huge panda (with a trunk), and Sakuma gradually turns into a panda when Behemoth doesn't accept her sacrifice.
Our Angels Are Different: They can't interfere with human affairs unless inflicted by a Demon, and can't use their angelic powers to fight unless they need to. They have wings, but only use them to ascend to heaven, and are charged with finding grimoires in the human world. If an angel takes a grimoire up to Heaven, the demon dies, permanently. The angels currently on earth appear to have a time limit set on finding grimoires, and up above there seem to be 'angel families' (Zeruel has a mother and two cats, for example) and apartment blocks. If an angel fails in getting a grimoire, see Mortality Ensues.
Rock Me, Asmodeus!: Lucifer tries to demonstrate his awesome guitar-playing skills to Akutabe, but his cute koala paws make it difficult. (He also wears leather pants and sunglasses, and has the ego of a rock star.)
Ship Tease: Akutabe and Sakuma, especially in episode 9 when Akutabe asks Beelzebub and Azazel to go see what Sakuma's up to when her work habits change.
In the manga Azazel implies that Akutabe just pretends to have a hands-off approach to Sakuma learning to be a demon summoner, and is actually watching her from the shadows. (Then again, it's Azazel who says this so it may or may not be true.)
Shout Out: One of the segments from the ending of episode 10 mirrors the iconic fall of Queen Beryl.
Undine's powers turn Golgotha Middle School into something straight out of Fist of the North Star.
Stealth Pun: In universe, Moloch gets a show named after him, and it's noted in the anime how the merchandising has gone through the roof with popularity around his character. Get it? He's become a Cash Cow Franchise.
Because of this he is the least popular demon in the manga among female readers. One of the reasons why the director of the anime gave the role to Kazuya Nakai was to boost Salamander's popularity. It worked.
He's so bad, the first thing he does to Sakuma when she greets him is spit on her face. Later, when the man who summoned him introduces himself to her, Salamander chastises him.
Stringy-Haired Ghost Girl: Sakuma claimed to be able to see one after Undine had "cured" her myopia.
Sudden Humility: After spending most of his introduction being a thorough haughty Jerk Ass, Moloch is forced into obeying Akutabe's laws. To wit: He gives a polite 'Itadakimasu', and eats his own kind (Bull demon forced to eat beef). He then proceeds to cry all the way back to the Demon world.
Throw the Book at Them: Akutabe is fond of doing this to his demons, with the added effect that the books he uses are grimoires, and touching a grimoire has very painful consequences for demons.