It depends on the different campaigns I've run and it's evolved over time. The way I set up the last campaign, my PCs were just a bunch of people who were at the same harvest festival in a big town when a murder occurred very close to them. Since they were all witnesses to the crime and saw the murderer, and being the 'heroic type,' I suppose they all went after the guy and caught him. That's how I got them to meet, and how I got them involved in the story of my adventure.
But were they heroes, ultimately? Well, they did heroic things but not always for positive reasons. One was an evil cleric infiltrating the church of a good God, another was a half-orc conflicted between his human and bestial natures, a third was a sociopathic thief with no compunction about committing murder if need be (or as the evil cleric called her "a girl with a problem." The others in the group were a bit less well defined or surefooted as players, so they don't count.
They didn't call themselves anything, they just did stuff for kicks, to prove their worth to their estranged step-father, for profit or as a means to an end (like bringing down the church of holiness and good). They did missions for the local Margrave because they were damn effective and he found he could make use of them, but they weren't formal "heroes," and there was no such thing as "adventurers" as a profession.
They once met a rival band in their area, kind of their mirror image: a bunch of religious fanatics, doing deeds for the benefit of the church. They hated them and would have tried to kill them, but it seemed their rivals were more powerful than they were, so they let it go. But again, motivation was something apart from just: let's go on adventure 'cause that's what people do. I like for there to be better reasons for doing these things than the standard tropes that have become way too cliched in gaming.
I was thinking of running a game that was all about courtly intrigue, next. All the PCs would be involved in the court of small kingdom in some way (some might be nobles while others might just be palace guards). They would be allied and working together, but I'm not sure you could call them an adventuring party. Trying to find a different way to do a campaign - something outside the norm of D&D gaming.