What is the Isur of "Molech"?
Rashi: "Molech" is the name of an Avodah-Zarah. 1
Ibn Ezra, Ramban (from Sanhedrin 64a): It is anything that people coronated over themselves.
Targum Yonasan #2, Tosfos Yom Tov (Megilah 4:9, citing Aruch (Arma'ah)): It is a prohibition against having Bi'ah with s Nochris to impregnate her to have a child that will serve idolatry. 2
Ibn Ezra, Ramban: Perhaps it is Malkom, the god of Amon. However, below (20:5) the Ramban proves that Molech is an irregular form of worship, which Amon called by that name because he was their king, but which applies equally to any Avodah-Zarah that one worships in this (irregular) way (See Ramban there DH 've'Hinei').
The Mishnah says that one who says so, we vehemently silence him! Tosfos Yom Tov - the Mishnah discusses one who limits the Isur to 'Aramiyusa', a particular nation that worships Molech. Really, it applies to Bi'ah with any Nochris, like Tana d'Vei R. Yishmael (25a) explains (refer to 18:21:7:2*). See also Na'ar Yonasan. (If it is among the Arayos, the Kares below (verse 29) should apply to it. Why did Sanhedrin 82a learn Kares only from Nevi'im (Mal'achi 2:11, "u'Va'al Bas El Nechar")? (PF)
What did the service of Molech entail?
Rashi: The father would hand over his son to the priests who, then arranged two rows of fire and passed the son between them by his legs. 1
Ramban (based on Yerushalmi Sanhedrin 7:10), Moshav Zekenim: The father would hand over his son to the priests, 2 who waved or brought him in front of Molech and handed him back to his father, who then puts him in the flame. Chachamim said that this is "Ma'avir Beno u'Vito ba'Esh" (Devarim 18:10). 3 The prophets of Ba'al would do so to (falsely) predict the future.
Targum Yonasan (20:2): The father would hand over his child to be burned in fire.
Did the child die?
Rashi implies that he did not die (he was merely passed between fires). Also Sanhedrin (64b) absolves someone who passes himself through the fire from punishment (Ramban).
Ramban, Moshav Zekenim: Yes, in most cases, he died. 1 We can say that one is liable once a limb catches fire; the service is completed when the child is totally burned.
Ramban: As the Pasuk in Yechezkel (23:37, 39) implies. See Ramban DH "ve'Im Kol Zeh", who elaborates at length. Indeed, the Gemara in Sanhedrin describes how Achaz passed his son Chizkiyahu to Molech, and it was only because his (Chizkiyahu's) mother smeared with salamander oil, that he survived (Ramban, DH v'Raboseinu). See also Ibn Ezra.
What are the ramifications of the word "mi'Zar'acha"?
Torah Temimah (citing Sanhedrin, 64b): The word "Zar'acha" insinuates that one is only Chayav for handing over one's children, but not one's father or mother; whereas the "Mem" teaches us that someone who hands over all his children is Patur.
According to Targum Yonason (refer to 18:21:1:3), perhaps it obligates even one who has a Yisrael wife, but he also has Bi'ah with a Nochris. (PF)
Why does the Torah use the double expression "Lo Siten ... Leha'avir la'Molech"?
Rashi: They refer to handing one's son over to the priests and passing him between the two rows of fire, respectively.
Ramban: The father hands over his child to the priests, takes him back, and puts him in the flame.
According to Targum Yonason (refer to 18:21:1:3), he has Bi'ah with a Nochris, and she becomes pregnant with a child that will serve idolatry.
What makes sacrificing to Molech a Chilul Hashem?
Ramban, Moshav Zekenim and Seforno: The Chilul Hashem lies in the fact that one sacrifices only animals to Hashem and one's sons to Molech. 1
As the Pasuk intimates in Yechezkel, 23:37, 38, 39 [Ramban]). It is also Metamei the Mikdash (as the Torah adds in 20:3), which refers either to the desecration of the sanctity of Yisrael, referred to below (11:34. "Metamei" is a term that the Torah uses when it speaks about major sins - See, for example, Amos, 2:7]), or to the desecration of the Beis-ha'Mikdash, when they sacrifice their children to Molech and then proceed to the Beis-ha'Mikdash to bring Korbanos (See 19:31, Yechezkel, 36:18 [Ramban]), or to the sanctity of Hashem, which is contained in his son (as it is contained in every Jew [See Ramban, DH 've'Al Derech ha'Emes]).
Why does the Torah put Molech amidst the Arayos?
Seforno: With the promise that Hashem made to Avraham (in Lech-L'cha) to be "his G-d and the G-d of his children after him", 1 the Torah taught the Parshah of Arayos, so that the children should be worthy for His Great Name to fall on them. It therefore follows with the Parshah of Molech, to make it clear that, sacrificing one's son in order to render one's other children worthy of serving Hashem - not only will it not succeed, but it is also a Chilul Hashem.
Yayin ha'Tov: This supports Targum Yonason and the Aruch, who say that the Isur is 2 to father a child from a Nochris.
Moshav Zekenim: It is put next to Eshes Ish, to teach that one who had Bi'ah with an Eshes Ish, in the end he will father a child who will pass himself to Molech and serve idolatry.
Hence the Pasuk concludes "Ani Hashem" - implying that Hashem has not changed, and that that promise remains intact (Seforno).
Tif'eres Yisrael (Megilah 4:9): Tana d'Vei R. Yishmael (25a) said 'the verse discusses Bi'ah with a Nochris.' I.e. it discusses both giving children to Molech (the simple meaning), and Bi'ah with any Nochris. (A support is that below (20:2-5), Targum Yonason implies that it discusses giving children to Molech. Rashi (25a) explained that R. Yishmael explains the mistaken Perush that our Mishnah censures. - PF)