In my definition a campaign is a multi-session D&D play experience possessing continuity.
Continuity is Required and can be of one or more of the following:
a) Of characters (the same character doing any number of things in the world, together or apart)
b) Of party, where PCs may come and go but the party remains
c) Of a defining campaign element--perhaps it is based around an everchanging and unconnected roster of PCs and parties attempting to defeat the evil empire
d) Of NPCs, which may serve as patrons, villains, or other connections for various groups of PCs
It Often Has, but does not require:
a) A guiding or defining plot
b) A theme
c) A consistent setting
Now, my impression of what a campaign is when I hear the word involves:
1. Multi-year play. If it isn't at least 2 or 3 years I have a hard time conceptualizing it as a real campaign.
2. Significant chunk of the characters' lives. If it's just a single adventure, I also have a hard time seeing it as a campaign.
3. More than a single story. Again, if it's just one long story, it's more of a long adventure than a campaign to me. A meta-plot that runs throughout the campaign is fine though.
My preferred style of campaign involves:
1. Many years of play. While it's still something I'm looking forward to, I get a twinge of envy when I heard grognards talk about their campaigns that are still running after 20+ years.
2. The entire adventuring life of the characters. While you might start at a higher level than 1st, once it gets going I expect a campaign to take characters to retirement (whether that means becoming rulers or gods, or just becoming old and passing their swords on to their grandkids).
3. A "Sim Adventurer/Hero" feel. I want a campaign to feel less like an attempt to tell a story, and more like characters carving out their way in the world. That doesn't mean they can't be heroes, or that there can't be a defining meta-plot, or even that a literary quality story can't evolve naturally, but what it does mean is that they aren't beholden to it, and can do whatever they want in a world that feels like it exists beyond their own choices (which is why I almost always make extensive use of random tables--so that I can be surprised as the DM too!)