During The Avengers, Black Widow mentions to Hawkeye about Budapest:
Black Widow: It's like Budapest all over again.
Hawkeye: You and I remember Budapest very differently.
What happened in Budapest? I'm sure it was a mission, but is this a nod to a previous story in a comic, or to something (perhaps) in production as it's not been seen yet?
It's a very frequently used and popular trope with writers:
The Noodle Incident is something from the past that is sometimes referred to but never explained (or, rarely, left conspicuously unexplained until a critical plot point), with the implication that it's just too ludicrous for words, and the reality that any explanation would fall short of audience expectations. Questions about it are often met with "You Don't Want To Know…"
Joss Whedon has used the concept in Buffy many times and the Avenger's movie is even referenced in the trope's Film sub page.
If you want an in universe explanation you'll have to ask Mr whedon and I'm sure he won't tell you.
This has been answered in the recently released Black Widow film. The spoilers will be unmarked, though they are only about the Budapest mission, as otherwise it will be too cumbersome. Proceed at your own risk.
Natasha remarks about her mission in Budapest to Yelena when they meet up. The mission was to kill Dreykov, the leader of the Red Room, and bring the Red Room down. They succeeded in the latter (from their perspective), however, mistakenly believed Dreykov was dead when he wasn't.
The remark about attacking the Chitauri being similar to Budapest is likely because in Budapest as part of the mission they had 10 days of shootout against waves of Hungarian Special Forces. This is likely at least somewhat similar to what we see in them facing off an unending force of waves of Chitauri.
Natasha: Dreykov’s dead. It took almost destroying the entire city just to get to him.
Yelena: If you’re so sure, then tell me what happened. Tell me exactly.
Natasha: We rigged bombs.
Yelena: Who’s “we”?
Natasha: Clint Barton. Killing Dreykov was the final step in my defection to S.H.I.E.L.D.
Yelena: Simple as that?
Natasha: Yeah, sure, “simple.” That’s what I’d call imploding a five-story building and then shooting it out with the Hungarian Special Forces. Took 10 days in hiding before we could even get out of Budapest.
Black Widow
As part of the film we see Nat and Yelena go to hide in a vent and Nat notes the below. So we know it wasn't particularly easy even for two top level agents such as those two.
Natasha: Barton and I spent two days hiding out up here.
Black Widow
