This is the approved revision of this page, as well as being the most recent.
There's a number of instances in which Hallel is supposed to be said. Please find the page relevant to the specific Hallel you are looking for.
- It is not said on Purim, see Customs of Purim.
- There is no Hallel on Rosh HaShanah and Yom Kippur because one should not sing on days when judgment is being rendered
- The proper place for the recital of Hallel is immediately after Shacharit. This is true even on a day that we recite Musaf and technically musaf should be recited first since it is said more often and tadir visheayno tadir, tadir kodem (the item said performed more often should precede the less common). 
- Hallel should be recited during the day and not at night. 
- Hallel should be recited while standing Some say that during hallel one shouldn't lean on a shtender in a way that one would fall if the shtender was removed. Someone who recited Hallel while sitting need not repeat it 
- An elderly person who has a hard time standing may recite Hallel while seated even lechatchila.
- If someone misses a word or sentence he must return to the place he omitted 
- In Hallel of Yom Tov one can interrupt just as one would for Kriyat Shema. That is, between paragraphs one can greet someone whom one has to respect and return a greeting to anyone. In the middle of paragraphs one can only greet someone whom one has to fear and return greetings to someone whom one has to respect.
- Gemara Arachin 10b, Rambam Hilchot Chanukah 3:6
- Sh"t Iggerot Moshe 3:70. Shulchan Aruch Harav 131:5 says that the reason is that zrizim makdimin limitzvot, that we are eager to fulfill mitzvot as early as possible. Sh"t Yaskil Avdi 8:20:55 writes that Hallel is a continuation of Yaaleh Viyavo and also that by saying Hallel in the place where Tachanun is usually recited, it is a proclamation of respect for this exalted day.
- Gemara Megilla 20b based on the pasul “From when the sun rises in the east until it sets, shall Hashem’s name be praised. ”
- Shulchan Aruch 422:7. Mishna Brura 422:28 says that we recite hallel standing since Hallel testifies to Hashem’s miracles and wondrous deeds, and testimony must be made while standing.
- Magen Avraham 422:11 says that one shouldn't lean on shtender in a way that one would fall if it was removed. Biur Halacha 422:7 s.v. "me'omed" questions this.
- Mishneh Brurahh 422:28, quoting Pri Megadim.
- Aruch Hashulchan 422:13
- Rama 422:6
- Gemara Brachot 14a, Rosh Brachot 2:5, Shulchan Aruch OC 644:1