What are examples of milchemet chovah and milchemet mitzvah?

Answer 1

There are three clearly defined Milchemet Mitzvah, required war, (מלחמת מצוות) in the Torah. One, Amalek, "you shall wipe out their name" - Deuteronomy 25:17. Two, the seven nations of Caanan, "you shall not allow any person to live" - Deu 20:16. Third, the nations of Ammon and Moab. Each case has a different set of principles of how to conduct the war, but in all cases it is required that one offer peace on the condition that they keep the seven Noahide laws (Worship one God, don't practice idolitry...) - Rambam, Laws of Kings 5:1. Now, if by Milchemet Chovah you are referring to the Kesef Mishnah 6:1 on splitting the category of Milchemet Mitzvah into two, the clear example would be that of a defensive war. This would be any war where one was attacked by an enemy or where ones peoples were targeted in hostility. Otherwise, the only other category of war would be that of a "permissible war" מלחמת רושות. These are any wars which are not defensive or required by Torah law, mainly for the sake of expanding land, wealth, or prestige. These wars are given a clear set of laws in order to declare them, and a very strict set of rules of engagement (i.e. not harming any none combatants or property).