The Mishna in Meg (6b) has a rule: אין בין אדר הראשון לאדר השני אלא קריאת המגילה ומתנות לאביונים. Therefore, this year, which is a leap year, we will not be reading megillah this month, nor can one fulfill the mitzvah of matanot la'evyonim this month.
Is there any mitzvah to have a party on 14 Adar this month? On the one hand, the Mishna only speaks about megillah and matanot l'evyonim, implying that when it comes to partying, there is no difference between the months. On the other hand, the gemara writes
ומתני' חסורי מיחסרא והכי קתני אין בין ארבעה עשר שבאדר הראשון לי"ד שבאדר השני אלא מקרא מגילה ומתנות הא לענין הספד ותענית זה וזה שוין
implying that it's only with respect to not having hesped or taanis that the days in both months are equal. Tos writes:
יש שנוהגין לעשות ימי משתה ושמחה בארבעה עשר ובחמשה עשר של אדר הראשון וריהטא דמתניתין נמי משמע כן מדקאמר אלא מקרא מגילה (ב) בלבד ומתנות עניים מכלל דלענין משתה ושמחה זה וזה שוין ולא נהירא דהא אמרינן בגמרא הא לענין הספד ותענית זה וזה שוין מכלל דשמחה ומשתה ליכא דע"כ לא תליא הא בהא דאי תליא הא בהא לאשמעינן דמשתה ושמחה נהגו בהו וממילא נאסר בהספד דהא הימים האמורים במגילת תענית האסורים בהספד אין בהן משתה ושמחה וכן הלכה שאין צריך להחמיר לעשות משתה ושמחה באדר הראשון
Worth noting that Tos speaks about the 14th and the 15th, but the gemara spoke only about the 14th, אין בין ארבעה עשר שבאדר הראשון לי"ד שבאדר השני. The Beis Yosef quotes from SM"K that even if you accept the notion that there is a chiyuv of mishteh in Adar I, it is limited to the day of the 14th alone.
והגהות מיימונית כתבו בשם סמ"ק שהעולם לא נהגו לעשות משתה ושמחה כי אם בי"ד שבאדר הראשון אע"פ שבאדר השני נהגו גם בט"ו שמחה ומשתה וטעמא משום דאמר תלמודא אין בין י"ד שבאדר הראשון וכו' ולא הזכיר ט"ו ושמא נקט אותו שהוא עיקר
(Perhaps the SM"K held that celebrating 14 vs 15 Adar are not two separate takanos, but are two opportunities for the kiyum mitzvah. See last year's post . Purim katan commemorates the source takanah, not the opportunities for kiyum hamitzvah.)
The SA and Rama (697) write that on Purim Katan, both 14 and 15 Adar I, we skip tachanun, we skip the mizmor of yaancha Hashem b'yom tzarah, and Rama paskens to avoid hesped and taanis. Rama then quotes the view that holds one should have a party of 14 and 15 of Adar I, and writes that even though it is not accepted l'halacha, one should try to have a larger seudah to be yotzei this opinion.
The Ksav Sofer al haTorah goes a step further with an even bigger chiddush (see also here). Tos only mentions having a seudah, but nothing about mishloach manos. Ran writes אבל לענין לשלוח מנות כיון דדמו למתנות לאביונים דליתנהו אלא בשני משמע נמי שאף משלוח מנות אינו אלא בשני. K.S. disagrees, and writes that since the point of mishloach manos is to enhance the seudah, if there is a chiyuv seudah in Adar I, there should also be a chiyuv mishloach manos as well. (See as well K.S. in the shut Y.D. 136 for a different approach).
R' Chaim Kanievsky goes yet a step further with an even bigger chiddush. This year 15 Adar I falls out on Shabbos. When 15 Adar II falls on Shabbos, you have Purim meshulash, and according to many shitos, the seudah is pushed until Sunday. RC"K therefore held that on 16 Adar I, Sunday, no tachanun is recited. Had it been Adar II, we would be celebrating on 16; therefore, we should celebrate on Adar I as well.
RC"K seems to be taking sides in a fundamental question. When we have a Purim meshulash and end up celebrating on the 16th, does that mean the 16th is the day on which we celebrate Purim, or is Purim really on the 15th, but since we can't have the seudah then (Yerushalmi says ain m'arvin simcha b'simcha), there is tashlumin on the 16th? If Purim is really the 15th and the 16th is just a day of tashlumin, it is harder to make the case that we should skip tachanun on that day even in Adar I. (In fact, it is hard to even make the case to skip tachanun or avoid hesped/taanis on that day in Adar II if you are not in a walled city. See here 5b)
The Rama ends off "tov lev mishteh tamid." Purim is our most joyous time of year. Shabbos is technically a day of oneg, not simcha. On Y"T there is simcha, but beis din used to send out policemen to make sure things don't get out of hand. The L. Rebbe notes that even on Purim itself, one has mitzvos ha'yom like mikra megillah that have to be taken seriously. The only day when one can be immersed almost completely in joy, he says, is Purim katan. This day is the pinnacle of simcha. Take advantage!