Somehow, the Blood War is the canonical D&D conflict that I am most familiar with--probably because of the 2e AD&D Monstrous Manual. The basic shape of the conflict is pretty simple: even though demons and devils are both principles of evil in the multiverse, they battle each other because they are aligned differently along the axis of chaos and law. There's a classic trope in there: evil fails because it can't help turning on itself.
The Blood War between the devils and demons is an eternal stalemate that is mostly confined to the Nine Hells and the Abyss. However, the Blood War does involve mortals. In fact, it goes a long way toward explaining why devils make pacts with mortals--granting them power in return for their souls. Hell needs soldiers, so forfeit souls generally become lemures or some other rank-and-file devil. Demons, on the other hand, have no use for errant souls, but they do encourage demonic cultists because even mortal fanatics can be used to disrupt diabolic plans.
Also, since both sides are locked in a war that neither can win--or even gain an upper hand--they scour the cosmos looking for artifacts, creatures, or...adventurers...who might be able to tip the balance and cause a decisive victory. Since most of the Blood War is fought in Avernus, the devils who die in battle are truly dead--unlike what happens when a devil dies on the Material Place. The demons who "die" are merely sent back to the Abyss, where they wander around like dummies, which effectively takes them out of the war.
The devils fight in the Blood War as a way of proving their superiority to the cosmos. They also see themselves as the heroes of the Blood War because they are keeping demons from overrunning the other planes. They're probably not wrong about that, actually.
The demons also fight in the Blood War as a way of proving their superiority to the cosmos. They also fight because they're really, really bored.
Another cosmic force--the agents of the Balance--step in to make sure that the devils never get the whip-hand over the demons and vice versa. Apparently Mordenkainen is one of those agents. They do things like stopping a crusade against deviltry because that would weaken the devils and possibly allow the demons to win the Blood War. It turns out that crying "both sides," as the agents of the Balance do, means that everyone else (rightfully) thinks they're douchebags.
Interestingly, there is no mention of Asmodeus, leader of the devils, being both a devil and a god--as he was in 4e. But we do get a story about how he was brought up on charges by a group of angels and asked that his case by judged by Primus, the Mucho Modron. Asmodeus argued that everything he did was by the book, and the angels kept heaping charge upon charge on him. Primus ruled that the angels were whiners, and that Asmodeus would have to carry the Ruby Rod--which forces him to be lawful. No, not this Ruby Rhod, unfortunately.
Zariel was one of the angels who brought Asmodeus up on charges. She was a bit of a Blood War fangirl, assigned to document the progress of the conflict. But she got drawn into the war, and took a contingent of mortals into battle against the devils. Asmodeus is a pretty decent CEO of Hell because he recognized initiative when he saw it and promoted her to archdevil status with a layer of the Nine Hells to personally manage.
Dispater is the arms dealer of the diabolic legions. He sends messages by carving them on the backs of imps, putting a little leather jacket on the imp to cover it up, and sending them on their way. Awww. But the leather jacket is linked to the imp's heart, so if anybody who isn't the intended recipient tries to take it off the imp dies and its body disintegrates before the message can be read. That's some diabolic Mission Impossible nonsense right there.
Mammon is a pretty generic greed-devil, but his minions have access to a book that gives the precise value of any soul. Nobody knows if Fierna and Belial are lovers, siblings, parent and child, or what--but it's clear that Fierna is the beauty and Belial is the brains of the operation. Their layer of Hell is home to the Diabolical Court, which is like People's Court but for devils.
Levistus rules over the cold part of the Hells, and he's trapped in ice kinda-sorta like the Satan at the end of Dante's Inferno. His job is to offer escape to ne'er-do-wells in need--such as a thief about to be hung from the gallows--in return for their souls. Since he's trapped in ice, he has a lot of time to really focus on his work. Glasya, Asmodeus's daughter, rules the prisoner layer of the Hells where those found guilty at the Diabolic Court are imprisoned and tortured. Glasya is quite the ingenue; she runs hell's crime syndicate.
Baalzebul once tried to usurp Asmodeus; as punishment he was cursed--if he lies to another devil he turns into a giant slug for a year. He finds this so embarrassing that he doesn't lie anymore. Oh shit, Mephistopheles also rules a cold layer of the Hells. (There's two of them??) Anyway, his layer is the magical laboratory of the Hells, and he hates distractions so much that he disintegrates his minions if they interrupt his thoughts. I get, Meph, I really do.