Campaigns are multiple scenarios with a common storyline, played against the computer. Most campaigns are single-player; while multiplayer campaigns are possible, that code is not yet well tested and no such campaigns ship with the game as yet. Once you have won a campaign scenario, you progress to the next. Keep in mind the following points:
The main difference between campaigns and individual scenarios is recalling units that survived previous scenarios. This is an important advantage that balances out later scenarios in a campaign being more difficult than early ones. Later in a campaign, you should usually have several high level units available to recall.
The importance of loyalty
Units with the loyal trait do not require any upkeep, regardless of their level. During a 20 turn scenario each level 3 unit will cost 80 gold to recall and maintain, but if that unit is Loyal it will cost only 20 gold. Multiply by the number of units you recall, and soon you are talking serious money! This is especially important in longer campaigns, where you will typically advance and recall many units.
Loyalty is hard to come by; the trait is only given to those units that join your cause of their own choice -- your leader, friendly units you discover hiding in villages, riders joining you, and so on. Keep them safe, level them up carefully, and they will be worth their weight in gold.
Go back and redo
If you are struggling in a campaign scenario, perhaps you should go back to an earlier scenario and try to win it more convincingly. If you lose all your high level units and your leader barely survives, or you use up all your gold, then later scenarios in the campaign may be very difficult to play. Note that some scenarios are designed to sap your army and gold, so this isn't a hard and fast rule.
Balance your army
Most campaigns call for a range of units. Especially when playing the campaign for the first time, you don't know what types of terrain or opponents you are going to face later in the campaign. Some scenarios are mostly open grassland, some are underground, some have lots of water, or lots of mountains. To succeed in completing a campaign, you will usually need to recall level 2 or 3 units of several different types. Keep this in mind when choosing what types of units to recruit. For instance, if your army has lots of mages, consider recruiting and levelling up some melee units in case you meet some Dark Adepts. If available, consider developing healers early in the campaign, you will probably need them later.
Once you have mastered a campaign, it may be fun going back and replaying it with an unbalanced army. For instance, some people have played most of HeirToTheThrone with only Shamans, Druids and Shydes. But doing this when playing the first time is probably just going to be frustrating and not much fun!
Exploit your last moves
When you are about to win a scenario, it is worthwhile maximizing the experience your units gain before you clinch the win. Normally you have to worry about counterattack by the enemy, but if you win the scenario the enemy won't have a chance to! So, on the last few turns, it is worth aggressively attacking enemy units as long as your own units do not die while attacking. On the last round, attack that Paladin with your Lich. You don't have to worry about the Lich being surrounded and killed next turn, since there will not be another turn.
Usually, it is worth trying to finish a scenario early. However, in some scenarios you may have gained control of all the villages on the map -- in this case it might be worth just collecting the income while milking the few remaining enemies for experience, instead of going for a quick kill.