For some years, Captain America and Iron Man were good friends. Perhaps not the sort of friends who would spend their days-off together, but quite chummy together as Avengers and frequently consulting each other about their problems as super heroes. Notably, when the Avengers fell apart in 2004 and were then rebuilt as a new team, Cap and Iron Man were the only name Avengers who returned to the team. Between Cap's leadership and Iron Man's finances, they remained the backbone of the Avengers. Until Civil War (2006).
Civil War began when the super-villain Nitro fought the New Warriors on a schoolyard and caused an explosion which killed hundreds of children (and the New Warriors). In the wake of this tragedy, some people demanded a higher degree of accountability for super heroes, which was not at all unreasonable - the Avengers had themselves been subject to oversight almost from their inception so ideally other super heroes would simply be expected to run their lives like the Avengers. Unfortunately, some people had an extreme hardline on this topic - notably S.H.I.E.L.D. director Maria Hill. It should be noted that the Maria Hill of the comics is nothing like her film counterpart - in the comics (especially in the era of Civil War) Hill would either a) harrass or b) obstruct super heroes whenever she met them.
As legislation for the Superhuman Registration Act was being mulled over, Hill summoned Captain America to a meeting for him to share intelligence with her about the super heroes. Under pressure, Cap admitted some heroes would surely oppose registration, making them true vigilantes. To this, Hill retorted, "So, nobody you can't handle?" Cap instantly took umbrage at the assumption he would help police an act he wasn't entirely in favour of, noting "Super heroes need to stay above that stuff or Washington starts telling us who the super-villains are," which was his basic same argument against returning to government service back in 1987. Hill then had her provactively-named Cape-Killers arm their weapons against Cap (yes, this legislation was not yet law and Hill was already preparing foot soldiers to murder super heroes). This finally turned Cap completely against her and he fought his way out of S.H.I.E.L.D.'s clutches, then started a resistance against the SHRA.
So do remember who started Civil War: it wasn't Iron Man; it wasn't Captain America. It was Maria Hill, because she's just the worst.
Iron Man took point on those heroes who supported the SHRA and battle lines began forming, but the SHRA passed midway through Civil War #2. Iron Man finally arranged a trap for Captain America's team in issue #3 so that he could attempt a peaceful solution (but first he tranquilized two of Cap's men - teleporters Wiccan & Cloak - so Cap had no escape route). Incensed at Tony for striking down two of his people, Cap attacked and both sides were committed in battle, until finally a cybernetic clone of Thor which Tony had created entered the battle scene and killed Goliath, long-time scientist, hero and best friend of Hank Pym. Cap and Iron Man's forces wouldn't fully meet again until the finale of Civil War in the brawl for it all, wherein Captain America surrendered after disheartedly seeing the public was on Iron Man's side. Cap would be killed while in S.H.I.E.L.D. custody en route to court. Tony Stark was then made director of S.H.I.E.L.D., which had the positive side effect of dethroning Maria Hill. Even though Hill had asked Tony to take her job for the gretaer good, his first act as director was to make her fetch coffee for him because Tony is a tiny, petty, twisted little nimrod.
Tony Stark joined Reed Richards and Hank Pym in devising various plans to improve the state of superhumans in the USA. Let's review what they accomplished!
- A super hero team in every US state! (were immediately infiltrated by agents of the Mandarin and Skrulls)
- A prison in the Negative Zone to incarcerate super heroes and super villains! (mixed prison population led to problems, basing the prison in the Negative Zone provoked attacks from Negative Zone's dominant powers)
- New super heroes created through Stark, Pym & Richards' technology! (super hero elitism run amock; screening processes still failed to locate infiltrators such as the aforementioned Mandarin agent)
- Thor was cloned! (and went on to kill many people)
- Norman Osborn was controlled by nanites in an assassination attempt on an Atlantean ambassador to intentionally provoke a war with Atlantis! (the war did not emerge but tensions remained hot)
- The Thunderbolts recruited super-villains into their ranks to become a state-sanctioned army of criminals! (rather than putting them in prison)
- Psychotic killer Norman Osborn was made director of the Thunderbolts! (positioning him to replace Stark as head of the Initiative)
In short, by codifying and militarizing all the superhuman might of the USA, Stark, Richards and Pym (or rather, a Skrull impersonating Pym) had simply made it all the easier for the government to abuse its power over ordinary citizens. Maybe anti-vaxxers have a point: you can't trust smart people.
Iron Man would eventually lose control of the Initiative after the events of Secret Invasion because Norman Osborn killed a Skrull, which proved that he was better suited to hold Iron Man's job (no, I don't get it either). While all the people who had trusted Tony and joined the Initiative found themselves completely under Osborn's thumb, Tony had bigger problems: looking out for himself. Er, wait, that sounds terrible. Tony had recorded all of the Initiative's data into his brain to keep it safe from Osborn, then wiped his own memory. Tony had his mind put back in order just in time for the crossover event Siege, but (oops!) Tony's back-up brain was copied prior to Civil War, so Tony didn't recall any of the terrible things he'd done.
But that's okay, because when he reviewed all of the data he concluded, "I am not sorry, and I'd do it all again." But hey, everything turned out fine - Captain America had even come back from the dead in a more-convoluted-than-usual-resurrection. In the end, no one was hurt. Oh, except for Goliath. And everyone killed under Osborn's watch. Any way, no one important was hurt.
And so I take my leave of you and hope you have enjoyed this brief look at Captain America & Iron Man's battles. I should perhaps mention they recently fought each other to the death in a story called Time Runs Out, but since that story wound up being written out-of-continuity almost immediately after publication, I don't think it matters.