There are Many Approaches
Ask any group of DMs, ever, and they will each have individual ways of making a campaign. I've listed just a few here. Other people here can and likely will post other methods.
The important part between having a string of adventures featuring recurring individuals and a campaign is the cohesiveness of it all. What you did last adventure affects this adventure. This can be achieved in a lot of ways.
Rule Book Guidelines
Very, very often, system rulebooks come with some suggestions for plots, and what to do as a DM. It seems most of them come down to figuring out a big idea for a plot, and breaking that into individual tasks the adventurers must complete. String these tasks together, and... it's a campaign!
S. John Ross's List
S. John Ross has a nice list of RPG plots here. While these plots are especially true of individual adventures, they can easily be broadened into a campaign. "Blackmail," "Clearing the Hex," "Don't Eat The Purple Ones," and "Pandora's Box" are all plots he lists that could be a starting point for a campaign.
Basically, you'd choose one as a very large plot, but then introduce sub-plots for individual gaming sessions which help solve the larger plot. This means what happened last session affects what happens this session. For most people, that's the key attribute of a campaign.
The Hero's Journey
It's just a theory, but Joseph Campbell came up with the idea that all (good) stories about heroes follow a certain pattern. If you modeled your campaign as a hero's journey, then your players will (most likely) feel pretty heroic and awesome when playing through it.
Some systems are better about this than others, but you could model after the Monomyth (Or Hero's Journey). Extra Credits did a good job of applying it to (video) games here, or you could check out TED's Hero Journey to get a good idea of what this is.
To apply this to roleplaying, you just need to make each segment or task in the hero's journey into an adventure. How will the adventurers experience the call to action? What or who are the threshold guardians? Where and whom do they gain more power from to destroy the shadow? Who or what becomes the shadow?
The Villain's Point of View
Some people like constructing a campaign by simply getting into the mind of their main villain (or BBEG), and figuring out what he/she does. Since the villain creates conflict, what is their overall strategy for accomplishing their goals? Boom, there's a plot! The heroes simply need to figure out what's up with that villain, and neutralize them.
This creates a kind of back-and-forth campaign, where the adventurers do one thing, and then the villain (or her/his lackeys) responds in kind. This allows for gradual buildups in the scale or intensity of the conflict, until they face BBEG and solve it once and for all.
Use Your Character's Motivations
This kinda feels like cheating sometimes, but just have people make characters, figure out what drives them, and make a campaign out of that. Do you have a kleptomaniac along with a guy who wants revenge on someone? Go steal all the hated person's earthly assets, and make an adventure out of that! Who cares about being creative when you can leverage the creativity of your players? I swear I read this (or something similar) on the Angry DM's Blog, which is just a good resource anyways.
Steal From Other Media
Take a plot from any film, book, or TV show you (ideally) really liked. This is more of a social no-no, especially when you get caught. It can still work really well, though, with minimal creativity on your part. Your players may love this or hate this. My players and I hate this sort of thing.
The added danger of this is that your characters may not generally react the same way as the characters in the original. Now you have a great plot that doesn't work anymore. This is why you steal the premise, not the actual entire plot.
The Conflict Approach
Get a bunch of cool (conflicting) situations, and throw them together in the same area. This becomes kinda sandbox-y, but makes for an interesting world! This way, there is a conflict, and characters' choices matter.
For example: A dragon wants to take over a nation and cultists want to summon an undead army (maybe as defense against the dragon?). Neither of these things have super great results, but it's a conflict. A conflict your characters can solve, one which will take multiple adventures to do so.
The other variant of this is summarized as follows: unreasonable responses for unreasonable situations. Create an unreasonable situation; clan x wants to destroy clan y, the king's daughter wants to become a dragon-queen, some wizard decided to toy with What-Ought-Not-Be-Toyed-With. Follow that to the (il)logical conclusion, which the adventurers being the ones to fix it. (The adventurers give the unreasonable responses, by the way.)
Choose a conflict. One that your characters care about. Diagram how they can solve it, and what needs to be done to do so. Adventures are just these "need to be done to solve the plot" things stuck together, plus whatever else you'll throw in for fun. Link the adventures, and that's a campaign.