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.
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.
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.
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.
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.
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.
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.
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.