II. What Does Shemuel Add?
Shemuel's statement is unique among all the other mishnayot in Pirkei Avot in that he merely cites a pasuk from Mishlei, without adding any explanation or commentary. Evidently, he felt that the theme contained in the pasuk was so crucial that it bore constant reiteration, and he would therefore cite this proverb often. The pasuk in Mishlei (24:17-18) reads: "When your enemy falters do not rejoice (bi-nfol oivekha al tismach), and when he stumbles do not feel glee (u-ve-koshlo al yagel libekha), lest Hashem notice and disapprove (pen yireh Hashem ve-ra be-einav), and avert His anger from him (ve-heishiv mei-alav apo)." This pasuk, which warns against triumphant celebration at our enemies' demise, raises numerous moral and theological questions. Perhaps the most famous application of this theme occurs on Pesach, when we refrain from reciting full hallel on the seventh day (and, by extension, during Chol Ha-mo'ed) since the Egyptians drowned on that day. As the gemara in Sanhedrin (39b) narrates, Hashem told the angels who wished to recite hallel at the time of keri'at Yam Suf, "My creatures are drowning at sea, and you will recite hallel to me?" Based on this gemara, the Shulchan Arukh (O.C. 490) rules that only half-hallel is recited after the first day of Pesach. (Recently, with the welcome death of Yasser Arafat - a murderer responsible for the brutal death of thousands of innocent Jews - this question resurfaced.)
This scripture talks about personal enemies, when 2 Jews fight, they shall not be happy with the other's fall. Regarding Israel and their enemies, it's the opposite, we're obligated to rejoice with their fall, because it's sanctifying G-D's Name.
Rabbi Kahane answers this here (hope you understand Hebrew):
http://www.rabbikahane.org/kahane/index.php?option=com_content&view=article&id=627:-q-18488&catid=45:2008-05-29-08-52-48&Itemid=129 He brings this Talmud (תלמוד מגילה דף טז עמוד את, Masechet Meggilah Daf 16 Page את) here:
"אמר ליה סק ורכב אמר ליה לא יכילנא דכחישא חילאי מימי תעניתא גחין וסליק כי סליק בעט ביה אמר ליה לא כתיב לכו (משלי כד) בנפל אויבך אל תשמח אמר ליה
הני מילי בישראל אבל בדידכו כתיב (דברים לג) ואתה על במותימו תדרוך"
I don't know how to translate it to English though, I think you may have studied it already.
Anyway, in general, we learn from it that after you've already destroyed them, humiliate them, by stepping on their stages (the stages back then were centers of Idol-worshipping and were the cultural symbol of many cultures, including the Persian one).
http://www.jlaw.com/Articles/war3.html
B. Seeking Peace Prior to Starting War
Two basic texts form Jewish law's understanding of the duties society must undertake before a battle may be fought. The Bible (Deuternomy 20:10) states:
When you approach a city to do battle with it you should call to it in peace. And if they respond in peace and they open the city to you, and all the people in the city shall pay taxes to you and be subservient. And if they do not make peace with you, you shall wage war with them and you may besiege them.
Thus the Bible clearly sets out the obligation to seek peace as a prelude to any military activity; absent the seeking of peace, the use of force in a war violates Jewish law. Although unstated in the text, it is apparent that while one need not engage in negotiations over the legitimacy of one's goals, one must explain what one is seeking through this military action and what military goals are (and are not) sought.35 Before this seeking of peace, battle is prohibited. Rabbi Jesse Hagalili is quoted as stating "How meritorious is peace? Even in a time of war one must initiate all activities with a request for peace"36 This procedural requirement is quite significant: it prevents the escalation of hostilities and allows both sides to rationally plan the cost of war and the virtues of peace.
Rabbi Shlomo Yitzchaki (Rashi), in his commentary on the Bible, indicates that the obligation to seek peace prior to firing the first shot is limited to Authorized wars. However, in Obligatory or Compulsory wars there is no obligation to seek a peaceful solution. Indeed, such a position can be found in the Sifri, one of oldest of the midrashic source books of Jewish law.37 Maimonides, in his classic code of Jewish law disagrees. He states:
One does not wage war with anyone in the world until one seeks peace with him. Thus is true both of authorized and obligatory wars, as it says [in the Bible] "when you approach a city to wage war, you must first call out for peace." If they respond positively and accept the seven Noachide commandments, one may not kill any of them and they shall pay tribute ...
Thus according to Maimonides the obligation to seek peace applies to all circumstances where war is to be waged. Such an approach is also agreed to in principle by Nachmanides.38
It is clear, however, according to both schools of thought, that in Authorized wars one must initially seek a negotiated settlement of the cause of the war (although, it is crucial to add, Jewish law does not require that each side compromise its claim, so as to reach a peaceful solution).39 Ancillary to this obligation is the need that the goal of the war be communicated to one's opponents. One must detail to one enemies the basic goals of the war, and what one seeks as a victory in this conflict.40 This allows one's opponents to evaluate the costs of the war and to seek a rationale peace. Peace must be genuinely sought before war may begin.
There is a fundamental secondary dispute present in this obligation. Maimonides requires that the peaceful surrender terms offered must include an acknowledgement of and agreement to follow the seven laws of Noah, which (Jewish law asserts) govern all members of the world and form the basic groundwork for moral behavior;41 part and parcel of the peace must be the imposition of ethical values on the defeated society. Nachmanides does not list that requirement as being necessary for the "peaceful" cessation of hostilities.42 He indicates that it is the military goals alone which determine whether peace terms are acceptable. According to Nachmanides, Jewish law would compel the "victor" to accept peace terms which include all of the victors' demands except the imposition of ethical values in the defeated society; Maimonides would reject that rule and permit war in those circumstances purely to impose ethical value in a non-ethical society.43
The Peace (Shalom) we offer is derived from the word "Hashlama" (acceptance) that they accept our rule without war, the peace is actually the surrender conditions (they accept the 7 Noahide laws, and be a tax-slave (I think it's translated to English this way; be a slave to the Kingdom, and pay a special tax, the Rambam explains this in Halachot Melchaim).
If they accept the conditions, there will be peace - and then we're forbbiden to touch the hair on the head. But if they don't accept even one condition, we never hear them until they accept them all. That's actually the Peace we offer, and when we come to a city we offer it to get out, make peace (according to our conditions) or war.
THESE RULES DO NOT APPLY TO SITUATIONS WITH MOAV, AMON (we don't offer them peace) AND AMALEK (the Arabs and the Muslims are Amalekites), with Amalek the very point of fighting them is to ELIMINATE them off the face of the earth in order to Sanctify G-D's Name.