Fare wares typically exist between airlines from different alliances.
What causes a fare war?
Fare wars tend to start as a regular sale by an airline for a group of flights to a region. A rival airline either matches or undersells the initial sale. This triggers a tit-for-tat approach with even other rival airlines joining in.
The most common fare war is for flights between the US mainland and Hawaii, in which United, Delta, American, Alaska and Hawaiian all compete to have the lowest price. This has been covered numerous times on Secret Flying.
What is an attritional fare war?
This refers to when an airline looks to dominate a specific market through lowering their prices and profits indefinitely. This is done in the hope that their rivals will not be able to take the same sacrifice of lowering profits in order to retain their demand in the market.
For example, Ryanair have aggressively pursued this in the past and continue to do so now. They are the most popular airline for intra-European flights despite being a low-cost carrier with sub-par services. The reason for their success and the decline of flag carriers in the intra-European flights market is simply because their prices are the most competitive.