A campaign is a linked series of battles which aims to recreate a sequence of events from the Warhammer World such as a Goblin invasion of the Empire, or an Undead excursion into the Dwarf realms of the Worlds Edge Mountains.
A popular form of campaign is to fight a fixed number of battles against the same opponent - four being a good number. The battles represent different types of battle: first a scouting encounter, then a raid, the third game is an ambush, and finally we have a big set-piece battle.
Scouting
The first battle represents a clash between scouting forces, so no war machines are allowed to take part. Alternatively, each player's choice of troops is restricted in some other mutually agreeable way.
Raid
The second battle represents a raid, with the player who lost the previous game fighting to defend a village, bridge, or some other suitable territory.
Ambush
The third game is an ambush, in which the player who lost the previous game tries to even things up by ambushing a column of enemy reinforcements.
Big Battle
The campaign culminates in a final battle. In this game each army receives a specific bonus or penalty which depends on whether it won or lost each of the previous games. The details can be agreed before the campaign begins. For example, you might allow each player an extra 250 points of troops for each game he has won. The player who wins the first game can place or move the scenery, as he has clearly out-scouted his foe. Similarly, the loser of the ambush might be denied heavy supplies by limiting his number of war machines to one.
This way of linking games together really changes the way battles are fought, as players have to consider the long term effects of their actions rather than just the result of the battle itself. Players who are interested in this style of campaigning will find the Warhammer Campaigns series of supplements very useful. These provide a number of battles together with special rules to represent the events depicted, as well as card buildings where appropriate.