What was the purpose of the two parties that Esther made for Haman and Achashverosh (Esther 5:4–8 and chapter 7), if one could have done the job? In other words, what happened that she needed a second party?
When Esther entered Achashverosh's throne room, a place full of idols, the Divine Presence left her (Megillah 15b). She realized, then, that such a place is not suitable for a miracle to take place. So she was going to have to get Achashverosh someplace where none of these would be present in order to be successful in her mission; the one place in the palace where that would be the case would be in her own private apartments. So she invited Achashverosh there for a party, and (for reasons listed in the Gemara there) had to invite Haman too.
The problem was that Haman came wearing his favorite outfit, the one on which he had embroidered a design of an idol (Esther Rabbah 7:5). So that spoiled her plan, and she needed to try again.
The next day was the one where Haman ended up having to lead Mordechai in procession around the city. Naturally, he didn't want his "god" having to witness his humiliation! So he wore a different outfit. Now Esther seized her chance: almost as soon as he got home, covered in filth and thoroughly discouraged, "the king's attendants arrived and made him hurry to Esther's party" (6:14) — that was at her instigation, so that he wouldn't have time to change into the outfit with the embroidered images. And so indeed the second party was free of idols, and Esther felt free to make her plea, which indeed was successful.
Some of the various reasons given in the g'mara (M'gila 15 amud 2) for Ester's inviting Haman — such as to make sure the Jews not depend on her being their friend in high places and cease praying, to appear to be befriending Haman so as to get him killed, and that pride comes before a fall — are strengthened by her giving two parties rather than one. (No source.)
The Tora T'mima (note 16, to chapter 5 verse 4) gives, in the G'ra's name, something that may perhaps also serve as a reason she wanted to put off her revelation a day: she wanted him handy at sof nidasah, since that is a good time for instigating an argument between Haman and Achashverosh. (However, it's clear from there that she knew of this reason when she asked Mord'chay to fast, so I don't know why she couldn't put off the fast a day and have only the "second" party.) (No source for its being a possible answer to this question.)
I saw a number of additional reasons mentioned in R Yosef Deutsch's Let my nation live (pp. 250ff)
- Esther was taking a cue from Jewish history. Yaakov prepared to defend himself against Esau in three stages (see Rashi on Bereshit 32:9: prayer, gifts and battle). Esther began with prayer, the first feast was a gift, the second feast would be the battle (R Meir Armah, Yalkut Shimoni, Menos HaLevi, R Shmuel di Uzidah)
- Taking another cue from history, where Yehoshua prepared to battle Amalek (= Haman) at Refidim tomorrow (Shemot 17:9). Apparently the battle with Amalek could only be won after a day's delay (R Meir Armah, Yalkut Shimoni, Menos HaLevi, R Shmuel di Uzidah)
- Esther knew Hashem does not let anyone suffer more than three days. By the next night she would have been fasting for three days and she was sure Hashem would guide her to success (R Meir Armah, Yalkut Shimoni, Menos HaLevi, R Shmuel di Uzidah)
- Esther's instincts told her the time was not yet ripe. The first banquet had been an ice-breaker but hadn't accomplished everything she wanted. Haman had warmed to her a little, but he was still a dangerous enemy, who could turn on her in a moment and convince the king to kill her. He had not yet let his guard down fully so that he would be vulnerable to her attack. She needed to shock him to his very core (Targum Sheni)
- Esther was concerned that Achashverosh would prosecute her for breaking the law [by entering to see him without permission]. She wanted to allow more time to elapse and distance herself from the incident (Akeidah)
- Haman did not yet feel fully equal to Achashverosh at the banquet, and therefore, Achashverosh's jealousy was not sufficiently aroused. The first banquet had been in honor of Achashverosh, with Haman attending in a subordinate role. The second would be given in honor of both Achashverosh and Haman equally, swelling Haman's head and discomfiting Achashverosh (R Yosef ibn Yachia, Midrash Lekach Tov, Yosef Lekach, Yedei Moshe, Menos HaLevi)
- Because Esther was fasting at the first banquet, she had not been able to join in the spirit of revelry, but at the second she would eat and drink with them. She would thus lull Haman into thinking her feelings towards him were warm, thereby inviting Achashverosh's jealous rage (R Yosef ibn Yachia, Midrash Lekach Tov, Yosef Lekach, Yedei Moshe, Menos HaLevi)
Simplest answer: it builds a lot more suspense.
The Vilna Gaon (to Esther 5:8) explains that Esther was worried that Hashem was not interested in saving the Jews at this time, and therefore her entire mission would inevitably fail. So she wanted to delay another day before revealing herself to see if perhaps she would get a Divine sign that Heaven was looking at them favorably. The following night, Haman was instructed to lead Mordechai through the streets in royal garb, which was carried out the following morning. Having received her sign, Esther went ahead and revealed herself at the following party.
Here Rabbi Yehoshua Zitron says from Midrash Meam loez that, The first time king Achsverous said, ".... Except for the jewish temple, I will not rebuilt it". Because the king added that, Ester thought that it is still not the right time yet to tell him about the whole situation.