A narrative campaign is a series of battles linked together by a story. This story is often created by an independent referee who can introduce special scenarios, victory conditions, strange events, extra rules and so forth. The campaign continues like this until the narrative suggests an ending.
For example, many years ago one of the authors ran a narrative campaign based on an Orc invasion of the Empire. The first game took place near the border, as an Empire army attempted to stop the Orcs before they penetrated too far into Empire territory.
The Empire army was decisively defeated and the referee decided that the second game would be about a small Empire force defending a vital bridge in order to slow down the Orc army's progress long enough for a new Empire army to be gathered together. A number of special rules and special pieces of terrain were used in this scenario to represent the unique situation. It was also a good example of an 'unbalanced game' The Empire army was horribly outnumbered, and couldn't hope to win the scenario in the sense of defeating the Orc army.
This didn't matter, however, as the Empire army's objective was only to slow the Orcs down, not to destroy them. The result was an interesting game that was very different from a normal run-of-the-mill battle. You'll be happy to learn that the Orcs were slowed down long enough for a large Empire army to be gathered to confront the Orc horde, setting up a large multi-player battle as the next 'episode' in the campaign.
In some cases, the result of a battle may suggest that one side or the other has a number of different options as to what to do next. For example, a victorious army might be within striking distance of two or more of its objectives, or a defending army might have to decide whether to counter-attack at once or wait for reinforcements. Considering such circumstances, rather than the referee deciding arbitrarily what to do next, the players could decide what happens. What the players opt for will alter the course of the campaign so that their strategic decisions, in combination with the results of the battles fought, will decide the outcome of the campaign.
Of course, a referee isn't essential in a narrative campaign - the players can discuss the various options and decide on the course of their actions between themselves. Alternately the players can take it in turn to devise the scenarios, with the referee running any scenarios that they create. The only important thing to remember is that a narrative campaign, more than any other type of campaign, is about more than just winning games - it's an exciting story where the results of your battles help decide the outcome.