The Source and Reason for Hallel
Hallel is recited on numerous occasions: on holidays and Rosh Chodesh, during the sacrifice of a korban pesach (Pesachim 64a), during the wine libations of regular sacrifices (Arakhin 11a), and when Jerusalem or the Temple was expanded (Shavuot 14b). What is the source for Hallel, and can differences be drawn between its various forms?
The gemara in Arakhin derives the obligation of singing Hallel while offering holiday sacrifices from several different verses. It is unquestionable that this form of Hallel is de-oraita (of biblical authority). The Rambam, however, rules that saying Hallel on holidays outside the context of sacrifices is only a rabbinic obligation. Support for this position may be found in a gemara in Berakhot (14a), which explores the issue of interrupting the performance of different mitzvot by talking. The gemara wonders whether an interruption (hefsek) would invalidate the recitation Hallel and megilla - EACH OF WHICH IS ONLY A MITZVA DE-RABANAN. Furthermore, as the psalms that comprise the Hallel were authored by King David, it would be difficult to envision them as de-oraita. The Rambam asserts this position in Mishneh Torah (Hilkhot Chanuka 3:6), and in his Sefer ha-Mitzvot (shoresh 1) he contests the Behag's ruling that Hallel is indeed de-oraita.
In truth, the issue he raises - that Hallel cannot be de-oraita since King David authored these texts can be easily resolved. As the Ramban notes (in his hasagot to the Rambam's Sefer ha-Mitzvot), it is quite possible that the concept of saying Hallel is a mitzva de-oraita, while the precise texts and timing of the mitzva were instituted by the sages. After all, the Rambam follows this strategy regarding the mitzva of prayer, which he believes to be de-oraita in origin (ironically, against the position of the Ramban), but concedes that the precise liturgy and timing were added only later by the sages. With regard to Hallel, however, the Rambam refuses to apply this reasoning and views the entire mitzva as purely rabbinic.
The Ramban defends the position of the Behag that Hallel is de-oraita. However, he lacks any direct verse obligating the recitation of Hallel. The only direct reference to such a practice is found in a verse in Yeshayahu which predicts the victory over Sancheiriv by declaring, "The song [after the victory] will be equivalent to the type sung on a night sanctified as a holiday [presumably a reference to yom tov]." In fact, the gemara in Arakhin employs this verse to distinguish between days on which Hallel is recited and days which do not require Hallel because they have no sanctity (such as Rosh Chodesh, which features no prohibition of work and hence does require a complete Hallel) or no status as a "festival" (such as Shabbat). In fact, the Ra'avad (in his hasagot to the Rambam, Hilkhot Chanuka 3:6) cites this verse in suggesting that Hallel is not a standard de-rabanan, and should be classified instead under the category of "divrei sofrim" (mitzvot which have a reference in Tanakh). The Ramban, however, cites no verse in the Torah to serve as the basis for the Biblical obligation of reciting Hallel.
The Ramban therefore claims that Hallel might indeed be a "halakha le-moshe mi-Sinai" - a mitzva which has de-oraita status even though no specific verse refers to it. Subsequently, the Ramban generates an source for Hallel. Every festival obligates us in the mitzva of simcha (rejoicing), based on the verse, "Ve-samachta be-chagecha, You shall rejoice in your festival" (Devarim 16:14). Generally, this simcha is actualized through sacrificial offerings, meat and wine, and, according to the Rambam, any other personal enjoyment (see Hilkhot Yom Tov, ch. 6). According to the Ramban, an additional expression of simcha is the recitation of Hallel. After all, the gemara in Arakhin (11a) had already determined that Hallel is the epitome of an "avoda" (service) which causes joy. If so, it stands to reason that this expression of joy should be incorporated into every Yom Tov.
We should note that the gemara in Arakhin that the Ramban adopts as his source merely establishes Hallel as a form of AVODA which causes joy. Hence, when sacrifices are offered on a festival, Hallel is required (as stated above). The Ramban extrapolates from here that all forms of Hallel - even those recited outside the Temple and the context of sacrifices constitute an expression of joy and are obligatory on festivals. This is not necessarily the implication of the gemara.
Another possible source appears in the gemara in Pesachim (117a), which suggests that Moshe and the Jewish people actually recited Hallel (in addition to the "Song of the Sea") when they crossed the Red Sea. This would support the Ramban's contention that Hallel traces back to Moshe Rabbeinu. An additional gemara which supports the Ramban's position is found in Ta'anit (28b), claiming that Hallel on Rosh Chodesh (on which there is no prohibition of labor) is only of rabbinic origin. This would imply that other forms of Hallel - namely, its recitation on sacred days of festivals - may be viewed as de-oraita.
There is another form of Hallel that might have earlier roots, even according to the Rambam. The gemara in Pesachim (117a) claims that during the Exodus, the prophets instituted the practice of reciting Hallel any time a grave danger facing the Jewish People was relieved. Thus, during the performance of a miracle (perhaps only a national one), we have an obligation to recite Hallel. In fact, the Brisker Rov claimed that the annual Hallel recited Pesach night stems from this requirement. Since the mitzva of sippur yetziat Mitzrayim (recounting the Exodus) requires us to envision ourselves as if we are currently departing Egypt, we actually relive a miracle and must therefore recite Hallel.
The Brisker Rov assigns a different nature to the Hallel recitation of Pesach night. Whereas normally the mitzva entails READING (keri'a), in this instance it has the quality of song or poetry "shira." Women would therefore be obligated in this specific Hallel, even though they might not be obligated in classic Hallel, as it is a time-bound mitzva. Since this special Hallel involves a direct and immediate response to the miracle, we would apply the principle of "af hein hayu be-oto ha-nes," they too were part of the same miracle (see Tosafot Sukka 38a s.v. mi). No blessing would be recited (see the Ran in his comments to Arvei Pesachim), and an interruption might be tolerated (as we actually allow during Hallel on Pesach night). Clearly, this form of Hallel would constitute a mitzva de-oraita. Whether the sages can legislate this type of Hallel beyond the immediate moment in which the miracle was performed is itself debatable, and would greatly impact the status of Hallel on Chanuka, which is neither a festival (as defined by the Torah) nor sanctified by a prohibition on labor, yet obligates one to recite Hallel because of the miracle that occurred.