In Malchus there are 3 aspects.
Rambam [Sefer Hamitzvos 173] : 1] "יקבץ כל אומתינו" - Gather the entire nation. That is the עצם המלכות. The aspect of "מלך על כסאו" - King on his throne.
2] "ינהיגנו" - Lead us. That is the leadership that emerges from the kingship. Judgment. משפט ודין.
3] Rambam in Hilchos Melachim [4-10]: The purpose of the king is to "להילחם מלחמות השם" - wage wars of Hashem.
We see these three aspects regarding the opposition of Amalek
1] אין השם שלם - The Name isn't complete. Leadership.
2] ואין הכסא שלם - The throne. Basic עצם המלכות.
3] מלחמה לה' בעמלק מדור דור - War.
This means that Amalek opposes all 3 aspects of Kingship.
On Shabbos we say 3 prakim that start "ה' מלך" that reflect these aspects of Malchus:
1] "ה' מלך גאות לבש ... נכון כסאך מאז" - The כסא is mentioned and the entire perek talks about the עצם המלכות.
2] "ה' מלך ירגזו עמים... ועז מלך משפט אהב". The judgment.
3] "ה' מלך תגל הארץ ... אש לפניו תלך" - Rashi says on "ה' מלך" - the following: "בנטלו המלוכה מעמלק". And then on "אש לפניו תלך" - he says "במלחמת גוג ומגוג". The wars.
The Yalkut Shimoni on Tehillim says "אהבתי כי ישמע ה' את קולי תחנוני ובימי אקרא"
ובימי אקרא - with my days I will call [out to Hashem]. The Medrash says that this refers to 3 days - Shabbos, Rosh Hashana and Yom Kippur.
These 3 days ALSO reflect the 3 aspects of Malchus:
1] Shabbos is when Hashem is Melech on his throne. "אתאחדת כורסיא יקירא קדישא ברזא דאחד" - The unity of his honorable holy throne with the secret of one . עצם המלכות. As we say "ישמחו במלכותך שומרי שבת".
2] Rosh Hashana is when Hashem is the King who judges.
3] Yom Kippur is when He is King at war. This is the day when evil is eradicated. Satan takes the day off. The שעיר לעזאזל - scapegoat [representing the evil] is sent over the cliff.
בימי אקרא - On these three days there is a special reason to daven . These three days encompass the 3 areas of Malchus. [Dovid Ha-melech said "ואני תפילה". Tefillah is Malchus].
The Ran in Rosh Hashana [ג, א מדפי הרי"ף] asks according to the opinion that the world was created on Rosh Chodesh Nissan, why we are judged on Rosh Hashana of all days? He responds that on Rosh Hashana, Hashem started to be appeased by Moshe [after חטא העגל] and was totally appeased on Yom Kippur. So Hashem judges us on these days of כפרה and סליחה.
The Gemara says that Hashem normally gets "angry" for a split second and that's it. But when Bilaam wanted to curse us He didn't get angry at all. Tosfos asks what he could have said in this time period to curse us. Tosfos answer that he could have said "כלם" - destroy. Hashem transformed that into מלך [same letters] as it says "ותרועת מלך בו". On Rosh Hashana we are ממליך Hashem via תרועת השופר and thereby transform the curse into a bracha.
For more - here.
