It states (Esther 9:27) הימים שני את עושים להיות ושנה שנה בכל וכזמנם ככתבם האלה, "to make these two days...every year." There are two days of Purim, the fourteenth and the fifteenth of Adar, and sefarim write that even those who celebrate Purim on the fourteenth should celebrate on the fifteenth as well.
One year, Shushan Purim was on Friday, and Rebbe Yehoshua of Belz zt'l heard someone say, "Neh, it isn't really Purim today." To teach this person and his community the importance of Shushan Purim, he began kabbalas Shabbos by reciting (Tehillim 29) אלים בני 'לה הבו לדוד מזמור, as we do when Shabbos follows a yom tov. He wanted to show them that Shushan Purim is a yom tov. This is the minhag in Belz until today. When Shushan Purim is on Friday, kabbalas Shabbos begins with לדוד מזמור, so we should understand the importance of Shushan Purim.
Mishmeres Shalom writes, "Shushan Purim also has a great, wonderful light. Therefore, it is proper to be happy on this day and to make a seudah. תמיד משתה לב טוב, 'He who has a cheerful heart always has a feast' (Mishlei 15:15).
The Tzemech Tzaddik of Vizhnitz zt'l (Purim המנהג ה"ד) writes, "The custom is to be very joyful on Shushan Purim... more than Purim itself."
Yismach Yisrael (9) writes, "We saw many tzaddikim who were more joyous on Shushan Purim than on Purim."
The Minchas Elazar of Munkatch zt'l (Shaar Yissaschar, Yemei Sason 31) quotes the Tikunei Zohar (Hakdamah 13.) that says the two days of Purim, on the fourteenth and fifteenth of Adar, are alluded to in the pasuk that discusses the war against Amalek (Shemos 17:16) ה"י כס על ד"י כי. The second date, ה"י, the fifteenth, is Hashem's name. Therefore, it is understandable that it is a greater day.
Also, the Chasam Sofer zt'l (Drashos 5590) writes that, logically, the primary day should be the fifteenth "because how does it help if there is peace in the world but the war is still raging in the king's capital? Therefore, the primary day for celebration is the fifteenth."
The Rebbe of Sadigura teaches, "The primary Purim is on the fifteenth of Adar because Yerushalayim is in the center of the world... Also, on that day, the moon is full."
As we wrote above, the Ateres Tzvi of Ziditchov zt'l said that one could accomplish more with tefillos on Shushan Purim than on Purim itself.
The Gemara (Megillah 7:) says that at a Purim seudah, Rabba accidentally killed Reb Zeira. The next day, he resurrected him. The Imrei Noam zt'l explains that the next day was Shushan Purim. On that day, one can create miracles with his tefillos to the degree of techiyas hameisim.
Rebbe Mordechai Chernobyler zt'l (Likutei Torah, Parashas Zachor, כתיב ה"ד) writes, "Whoever has joy and a meal on Shushan Purim...it is certainly a great mitzvah, as it states (Esther 8:15) ושמחה צהלה שושן והעיר, 'The city of Shushan shouted and rejoiced.'" He says that this is alluded to in the pasuk (Shir HaShirim 5:13) שפתותיו עובר מור נוטפות שושנים, "His lips are like roses, dripping with flowing myrrh." He explains that שושנים alludes to Shushan Purim. Celebrations on this day עובר מור, are like besamim that give off a pleasant scent.
A freilechen Purim and Shushan Purim to all!