Answer:
Holy war, in the context of an inner struggle, is about a person's efforts to act as their religion teaches them they should. Some examples are a Christian praying for forgiveness if he sins, or a Muslim making her pilgrimage to Mecca.
Holy war comes from the trouble people have when they want to do the wrong thing but know that they should do that right thing. Sometimes this is difficult, I'm sure you can think of a time when you wanted to do something that you knew was wrong. When the thing that a person should do is defined religiously this internal conflict is holy war. In the above examples the Christian's right choice is to pray and wrong choice is to ignore the sin, the Muslims's right choice is to go to Mecca and the wrong choice is to not go despite having the means to do so.
Sometimes holy war extends to external forces, in these cases it is no longer an internal struggle. Holy war can be defensive, like when a person justifies her religious beliefs to a non-believer who has taunted her, or when a person has to decide how best to represent himself as a follower of his religion.
Holy war can also be offensive, in these cases it is not an internal struggle and some people disagree that these count as holy war. One example would be if somebody insists that you define right and wrong by her religious beliefs. Historically people have claimed military campaigns, like the crusades, as holy wars, which implies that those wars are condoned by a higher power. This is controversial because many religions have rules against killing, and so military holy war is often associated with small groups of extremists and rarely (if ever) agreed upon by all followers of a religion.