Not to state the obvious, but heroes mostly fight villains. After all, villains are the ones who wreak havoc and oppose the ideals of truth and justice. Sadly, things aren't always that simple.
Superheroes sometimes turn against each other. They can be under mind control, or they could simply have different worldviews. At the end of the day, though, the real reason for these brawls is good old-fashioned fan service. Audiences love to see their favorite characters go at it. DC, in particular, has fulfilled that desire with numerous heroic clashes. At a certain point, one wonders how heroic these figures really are.
6 Roulette's Arena
This twisted villainess specializes in comic book bloodsport. Roulette routinely brainwashes metahumans and forces them to fight in underground gladiator arenas. Many of these unwilling combatants are members of the Justice League or other superhero teams. Some matches are one-on-one; others are battle royales. Either way, it makes for an interesting challenge.
That challenge lies not in overpowering the enemy, but in finding a way out of Roulette's gilded cage. The characters must fight hard enough to stay alive, yet not so much that they hurt their friends. Such a precarious balancing act is always a thrill to watch.
5 JLA/Avengers
It's always enticing to see top dogs go at it. Countless kids pit their action figures against each other, and adults still debate who would win among their favorite characters. DC and Marvel recognize that desire.
This fuels their line of crowd-pleasing crossover comics. Numerous members of both the Justice League and the Avengers compete for supremacy. These can be physical battles or friendly competitions. Yeah, they make for cheap thrills, but a thrill nonetheless. It's satisfying simply seeing such storied icons interact. Watching them fight is just a bonus.
4 Superman/Batman: Apocalypse
When Superman's impressionable cousin, Kara, arrives on Earth, multiple parties try to instruct her on how to use her powers. One such party is Darkseid, who makes the Kryptonian one of his Female Furies. She then attacks her cousin in a vicious rage.
Audiences naturally suspect brainwashing, but Kara's complaints seem oddly genuine. Superman and others have constantly sheltered the little Kryptonian, telling her who to be and how to act. That pressure would cause anyone to lash out, so fans are left guessing whether Darkseid is really controlling her. This makes the conflict more unsettling, as it potentially has no solution. The creators don't hold that tension for long, but the tale is effectively creepy while they do.
3 A Better World
The Justice League is the most powerful force on the planet. If they wanted, they could easily seize control from the world's governments. An episode of their TV show depicts an alternate universe where they do. When Lex Luthor becomes the US president and nearly starts a global catastrophe, Superman kills him. He and the rest of the League then impose their own rules on the people--rules which come at the cost of freedom. Viewers see how far these "Justice Lords" have fallen when they encounter their counterparts from the mainline universe. These teams obviously contrast in impactful ways.
What's more unsettling is how their goals mirror each other. The nobler Superman admits he's had the same temptations, meaning that he and the others are capable of the same actions. That message resonates long past their physical battle. It has chilling ramifications in subsequent episodes and later inspired the Injustice games.
2 Kingdom Come
This story presents a world in fear of metahumans. People don't trust the unchecked vigilantism inherent in superhero antics, thus igniting a war between the two sides. This puts Billy Batson in an awkward position. He's a human who can transform into a superhero named Captain Marvel (later known as "Shazam"), so he has a foot in both camps. Thanks to that, Lex Luthor brings him onto his side. This puts the magical crimefighter at odds with Superman, who is attempting to stop a bomb targeting metahumans. That scenario would be thrilling enough, but it pales next to the conclusion.
Batson ultimately sacrifices himself to stop the bomb. It's a bittersweet reminder that heroism can come from both humans and metahumans. That's cemented by the Justice League subsequently going public with their heroics. They become less vigilantes and more lawful agents, working hand-in-hand with the world powers. People often want transparency from those acting in their interests, so applying that thinking to superheroes is an intriguing turn. What's more, is that it feels earned after such a momentous act of selflessness.
1 The Dark Knight Returns
Once again, the distrust of metahumans forms the foundation of this tale. That distrust has forced the Justice League into retirement. Superman goes along to get along and becomes a government stooge. The president sends him after Batman once the Dark Knight resumes crimefighting. The resulting brawl is not just spectacle, but ideological.
The fight is tinged with conflict surrounding authority and how it's abused. It's especially subversive when that authority combines with human pettiness. These are the same ideas discussed in political debates about the government's role. True, the dialogue is somewhat on-the-nose, but it never becomes pretentious. Rather, it gives both heroes an understandable and timeless worldview. That's why this duel still resonates with fans.