In The Avengers, Black Widow and Hawkeye, alongside other Avengers were in a battle which I cannot remember. However, they were distinctly talking about the Budapest being similar to the battle they were currently in. What really happened in the Budapest?
As per this answer on SciFi.SE, this is an example of a Noodle Incident.
From this page on TVTropes;
"The Noodle Incident is something from the past that is sometimes referred to but never explained, with the implication that it's just too ludicrous for words - or perhaps too offensive for depiction - and the reality that any explanation would fall short of audience expectations.
The key to this trope is that the audience is left to imagine what happened based on vague hints or clues, with funny (or scary) results.
In fact, that page directly mentions the line in question from Avengers Assemble;
In Spy Fiction or similar tropes, the Noodle Incident is usually referred to by a place's name. "This is just like Budapest!" "Remember Helsinki?" "God, it's Rio de Janeiro all over again." Sometimes this is elaborated upon, but usually not.
So in short, we'll probably never know exactly what happened in Budapest, and it's likely that writer/director Joss Whedon doesn't either - it is simply a throwaway line that helps us infer a long history between Hawkeye and Black Widow, informing us of the dynamic they share without bogging us down in exposition.
Per the new Black Widow movie, the Budapest scenario has been somewhat clarified.
Budapest was where Hawkeye caught up with Natasha and brought her into SHIELD (as referenced in the Avengers movie). Some additional details are given (arrow holes in the walls of Natasha's Budapest flat), Dreykov (who runs the Red Room) and his daughter (the one referenced by Loki during the interrogation scene in Avengers) who were supposedly killed by Natasha during the completion of her defection. Hawkeye and Widow spent two days hiding in a vent inside a subway station while the Hungarian Police hunted for them.