The practice before Purim is to donate half of the monetary unit of one’s time and place (e.g., half a dollar, half a pound, etc.) in memory of the half-shekel that was donated in Adar to purchase animals for the communal sacrifices. The custom is to give three times this half unit (e.g., $1.50) because parshas Ki Sisa uses the word “terumah” (donation) three times. (Rama 694:1)
The money is donated to charity (Shaarei Teshuva 694:2). Some specifically send it to the poor of Israel. (Yosef Ometz 1089) Harav Yitzchak Yosef shlit”a writes that one should give the money to Yeshivos that are teaching young Torah scholars. (Yalkut Yosef Kitzur Shulchan Aruch page 770)
The common practice is for the shul/tzedaka pushka to supply three half dollars. Each person gives $1.50 to the tzedaka in exchange for the 3 half dollars. He then lifts up the three half dollars in order to acquire them. And finally he donates the three half dollars in order to perform the mitzvah of Machatzis Hashekel. Harav Yaakov Kametzky zt”l (Emes L’Yaakov page 160) is cited as saying that one should give a little more than $1.50 in exchange for the 3 half dollars. Since we don’t like to make exact change from tzedaka.
The poskim offer three opinions as to when one should perform the Machatzis Hashekel. A) The Rama writes that one performs the Machatzis Hashekel before Mincha on Taanis Esther. B) Harav Yaakov Emden zt”l and the Chasam Sofer maintained that it should be performed following Mincha on Taanis Esther. (See Netai Gavriel Purim Chapter 26 and Yismach Yisrael page 28 for view of Harav Elyashiv zt”l) C) While others perform it on Purim morning (Magen Avraham 694:2). It seems the common custom is to follow the first opinion of the Rama. (Mishnah Berurah 694:4, Kaf Hachaim 694:25, Yismach Yisrael page 28 view of Rav Chaim Kanievsky shlit”a)
If one did not perform the Machatzis Hashekel before Purim he should perform it as soon as possible during the month of Adar. According to Rav Chaim Kanievsky shlit”a b’dieved one should perform Machatzis Hashekel even after Adar (oral ruling cited in Yismach Yisrael page 28). Similarly, the Sefer Avnei Yashfei (1:133) writes that b’dieved if one forgot to perform the Machatzis Hashekel he can do so for 12 months after Purim (until the following Nissan).
One may not use Maaser money for the Machatzis Hashekel. If one chooses to give more than the required amount ($1.50), the excess may be given from Maaser money. (Yosef Ometz 1088)
The Rama writes that one is only obligated to give Machatzis Hashekel once one reaches the age of twenty. The Mishnah Berurah cites the view of the Tosafos Yom Tov that one is actually obligated once he reaches the age of thirteen. The Mishnah Berurah then explains that the common custom is for the father to give Machatzis Hashekel for his children even those below the age of thirteen and even for those in utero. He adds that once his father gives on the minor’s behalf, he is obligated to continue doing so.
There is a debate amongst the poskim whether women are obligated to perform Machatzis Hashekel. The common practice is that the husband performs Machatzis Hashekel for the women in his family. (Magen Avraham 694:3, Netai Gavriel 27, Halichos Shlomo Purim 18:9 and Shevet Halevi 7:183)