Gemara

3b (Mishnah): We take securities [for Machatzis ha'Shekel] from Leviyim, Yisraelim, converts, but not from women, slaves, or Ketanim.

Once a father began giving the half-Shekel on behalf of his Katan, he may not stop.

(Mishnah): Even though it was taught that we do not take a security from women, slaves and Ketanim, if they gave the half-Shekel, we accept it.

4a - Question: "... [If they gave, we accept it] connotes that we do not demand that women, slaves and Ketanim give. "We do not take a security" connotes that we demand from them!

Answer: We demand from a Katan only if he brought two hairs.

Rishonim

Rambam (Hilchos Shekalim 1:7): Women, children and slaves are exempt from Machatzis ha'Shekel. If they gave, we accept from them. If a father began giving for his minor son, he may not stop. He gives every year until the child matures and gives for himself.

Maharil (Hilchos Purim 4): At Minchah, when we go to the Beis ha'Keneses, we give Machatzis ha'Shekel and Ma'os Purim. One who gives his Shekel must also give Ma'os Purim, which are three halves of the [currency of the] country. I saw in Magentza rich people, and some poor people, gave Machatzis ha'Shekel. They saved it to support people alighting to Eretz Yisrael l'Shem Shamayim. Ma'os Purim they gave immediately to Tzedakah, like other Tzedakah. Where there is no Minyan, one may hold Ma'os Purim to give to wherever he wants.

Bartenura (DH Shuv): The father may not stop [giving for his son] once he began. If the father died, he gives for himself.

Tosfos Yom Tov (DH Shuv): The Rambam connotes that if the father died, the son need not give for himself until he matures, unlike the Bartenura. I do not understand the Bartenura. How can the father impose an obligation on his son? (A special tradition from Moshe from Sinai teaches that a father can make his son a Nazir.) Perhaps he discusses when he inherited from his father, and he holds that there is a lien on the property.

Tosfos (Megilah 21a DH ha'Korei): We read the Haftorah at Minchah on a Ta'anis because it says in it "Shimru Mishpat v'Asu Tzedakah", and the reward of a Ta'anis is the Tzedakah given towards evening. Therefore, it is proper to read it [close to] evening, after people gave Tzedakah.

Poskim

Rema (OC 694:1): Some say that one should give before Purim half the fixed coin in that time and place, to commemorate Machatzis ha'Shekel that they used to give in Adar.

Bi'ur Halachah (DH Yesh Omrim): It is not clear whether or not an Oni financed from Tzedakah is obligated to give Machatzis ha'Shekel.

Kaf ha'Chayim (19): The Rema says "some say" because the Rambam, Tur and Shulchan Aruch did not mention this. In any case, the custom is like the Rema says, to commemorate the half-Shekalim that they used to give in Adar for Avodas ha'Mikdash.

Kaf ha'Chayim (22): In some places they send the money to Chachamim of Eretz Yisrael, to commemorate Machatzis ha'Shekel, which was collected in Chutz la'Aretz and sent to Yerushalayim. In Eretz Yisrael, some give it to the Gabai for expenses of the Beis ha'Keneses, to commemorate the Mikdash, and some covertly give it to needy Chachamim.

Rema (ibid.): One should give it on Purim night before praying Minchah. This is the custom in these lands.

Gra (DH ume'Acher): See Berachos 6b (which says that reward of a Ta'anis is the Tzedakah - Damesek Eliezer).

Mishnah Berurah (4): The custom is to give Machatzis ha'Shekel before Minchah, and in the morning Ma'os Megilah [to the one who reads the Megilah for the Tzibur].

Magen Avraham (2): In our lands, the custom is to give in the morning, before Keri'as ha'Megilah.

Darchei Moshe (1): Mahari Birin says that even pregnant women give three halves. The Siman is "Zeh Yitnu Kol ha'Over" (we can read "Over" like Ubar (fetus)). Maharil says that we collect at Minchah of Ta'anis Esther, when people go to the Beis ha'Keneses. This is the custom, unlike the Agudah, who says in the name of Maseches Sofrim that we give before Parshas Zachor. The Agudah also says that one does not give for the sake of Kaparah, rather, voluntarily.

Magen Avraham (3): Women are exempt. Hagahos Minhagim (Hilchos Purim 49) wrote that women and children are obligated. I do not know his source. The Mishnah says that if a father began giving for his minor son, he may not stop.

Kaf ha'Chayim (27): I say that since it says "Lechaper Al Nafshoseichem", we give also for women and children to atone for them. In the days of the Mikdash, they used to accept from women and children, and all the more so nowadays that the money goes to Tzedakah.

Kaf ha'Chayim (25): Sidur ha'Rashash connotes that even if the fast is pushed back, e.g. Purim (the 14th) falls on Sunday and the fast is pushed back to Thursday, we give before Minchah on the fast, to join it to the fast and to atone.

Rema (ibid.): Some say that we give Machatzis ha'Shekel to Tzedakah in addition to three half coins. This is not the custom.

Note: Presumably, "Machatzis ha'Shekel" refers to the value of half a Shekel Kodesh of silver. This is a 10th of the value of Pidyon ha'Ben, which was about 80$ or 300 Israeli Shekalim at this writing (5774).