Shulchan Aruch, Siman 490, Se'if 4

"On all the days of chol hamoed, and on the last two days of Yom Tov, Hallel is recited with omissions, just as on Rosh Chodesh."
(Shulchan Aruch)

(We continue in this siman from se'if 4 onward; the first part of the siman, including the laws of Havdalah and the Torah reading, was learned earlier.)

What "Hallel with omissions" means

The language of the Rishonim: "With omissions" is the phrasing the Shulchan Aruch adopts, and it means that we skip two chapters of Hallel.
The popular term "half Hallel": It does not mean an exact half, only that a portion of Hallel is said.
The question itself: On the face of it, we might have expected a complete Hallel on every day of Pesach, just as we say it on chol hamoed Sukkos, and the Shulchan Aruch does not explain the reason.

The Mishnah Berurah's reason: the drowning of the Egyptians on the seventh day of Pesach

"Hallel is recited with omissions, because on the seventh day of Pesach the Egyptians drowned, and the Holy One, Blessed be He, said: The works of My hands are drowning in the sea and you are singing songs of praise before Me? And since on the seventh day we do not say it, therefore on chol hamoed too we do not say it, so that chol hamoed should not be greater than the last day of Yom Tov."
(Mishnah Berurah, s.k. 7)

The Midrash: On the seventh day of Pesach the Egyptians went down into the sea, and about this the Holy One, Blessed be He, said: "The works of My hands are drowning in the sea and you are singing songs of praise before Me?" That song of praise is Hallel.
How "no song is sung" is fulfilled: Saying part of Hallel is not considered a complete song of praise, so that is enough for it not to count as singing over the drowning of the Egyptians.
The reason for chol hamoed: It makes no sense for chol hamoed to be greater than Yom Tov, and therefore on chol hamoed as well we do not say a complete Hallel, so that people should not become confused and wonder how it can be that on Yom Tov we say it with omissions while on chol hamoed we say it in full.
A reservation: Some do not fully accept this reason, since one could simply explain to the public that the seventh day of Pesach is exceptional because of the drowning of the Egyptians, and even so the question would remain: what prevents saying a complete Hallel on chol hamoed?

The berachah over half Hallel

The Ashkenazi custom: A berachah is recited even over the abbreviated Hallel, on Rosh Chodesh and on chol hamoed: "Who sanctified us with His commandments and commanded us regarding the reading of Hallel."
The Sephardi custom: No berachah is recited, and the text of the berachah they have is "to complete the Hallel."
The basis of the difference: According to the view that no berachah is made, a reading with omissions is not considered Hallel at all but merely the recitation of chapters of Tehillim, and one cannot make a berachah on "the reading of Hallel" when one is not reading Hallel in its entirety. And just as no berachah is made on Rosh Chodesh, so too none is made on chol hamoed.

The Gemara's reason in Arachin: the difference in the korbanos

The korbanos of Sukkos: Each day a different number of bulls is offered: on the first day thirteen, then twelve, then eleven. The change in the korban teaches that every day of chol hamoed Sukkos stands on its own and is not subordinate to the day before it.
The korbanos of Pesach: The korbanos are identical all seven days, as the verse says: "Like these shall you offer daily, for seven days." From here we learn that the days of chol hamoed Pesach are a continuation of the first day, subordinate to it and drawn along after it.
The conclusion: The first day is Yom Tov, and on it we say Hallel without any question. The days of chol hamoed, since they are drawn after it and do not stand on their own, do not have a complete Hallel; nevertheless, since they do have something of the festival about them, a portion of Hallel is said.
A proof from the text of Mussaf: The Mussaf prayer was composed according to the korban. On Sukkos the siddur spells out "on the second day," "on the third day," and so on, because the korban changes from day to day. On Pesach, by contrast, the text is the same on every day, and the only difference is the opening "and on the fifteenth day of this month," which is said on the first day.

The seventh day of Pesach and its link to the first day

The difficulty: The seventh day of Pesach is Yom Tov, and seemingly its status is higher than chol hamoed.
No Shehecheyanu: On the seventh day of Pesach we do not say Shehecheyanu, because the Shehecheyanu of the first day covers it. This shows that it too is bound and attached to the first day and is not an independent festival.
Yet it is still Yom Tov: Work is forbidden on it, and it does not have the leniency of davar ha'aved (a loss that would otherwise be incurred) that chol hamoed has. But since its korban does not change, on it too we do not say a complete Hallel.

Shulchan Aruch, Se'if 5: the mnemonic for the readings of the eight days of Pesach

"The mnemonic for the Torah readings of the eight days of Pesach: Meshoch, Tora, Kadesh, Bichaspa, Pesal, Bamidbara, Shelach, Buchra."
(Shulchan Aruch)

What the mnemonic is: Four words times two, eight readings, corresponding to the number of days of Pesach outside of Eretz Yisrael, including the final day of Yom Tov.
Why a mnemonic is needed on Pesach: On Pesach a different portion is read each day, unlike Sukkos where the reading stays in the same parshah and only the verses of that particular day change. This requires preparing the Torah scrolls and makes things difficult for the gabbaim and for those arranging the reading.
The words spelled out: Meshoch is "Draw out and take for yourselves lambs for your families and slaughter the Pesach offering," for the first day; Tora means "an ox"; Kadesh is "Sanctify to Me every firstborn"; Bichaspa is "If you lend money"; Pesal is "Carve for yourself two tablets of stone"; Bamidbara is the portion in the wilderness of Sinai, the parshah of Pesach Sheni; and after them come Shelach and Buchra.

The way of mnemonics in halachah

Why mnemonics are needed: Just as the mnemonic for the Seder night was established, "Kadesh, Urchatz, Karpas, Yachatz," because a person has to carry out an order of ten or fifteen actions and remember what comes before what, so too mnemonics were established for anything complex.
The historical background: Before printing, not everyone owned a Haggadah, and writing one or buying one cost a great deal of money. Many people said the Seder by heart, and therefore the mnemonics were indispensable.
The plain meaning of the mnemonic: Halachically the meaning of the mnemonic is straightforward: Kadesh is Kiddush, Urchatz is the washing of the hands before Karpas, and so on, and all of this is spelled out in the Haggados.
The proliferation of interpretations: Since the mnemonic was accepted throughout all of Israel, hundreds of commentaries have been written on it, much like the many commentaries on the Haggadah itself, including full derashos going in entirely different directions, for instance that a person who wishes to sanctify himself must wash and immerse. Clearly the author of the mnemonic did not intend any of that, but only the plain halachic meaning.
The general principle here: Something accepted by the entire Jewish people takes on a sanctity, and it is said that there is something in it and that it merited ruach hakodesh, and therefore commentators throughout the generations searched in it for additional hints. Derashos were expounded on the mnemonic for the readings of the eight days of Pesach as well, and their explanation will follow.

In passing · Wearing tefillin on chol hamoed

There is no explicit source: The Gemara does not address the question of whether tefillin are worn on chol hamoed. Those who dispute and debate the issue bring hints and nuances from the Gemara in both directions, and since the matter is not explicit it is hard to decide it from the sugyos.
The approach of the Beis Yosef: In the Beis Yosef, which is the lengthy commentary on the Tur and is something of a halachic encyclopedia, it initially appears that he leans toward wearing tefillin, and in the end he accepts the words of the Zohar that tefillin are not worn on chol hamoed.
The place of the Zohar in tefillah: The Zohar has a serious role in the laws of prayer, not only by way of the Arizal but even before him, and it contains references to these matters.
Outside of Eretz Yisrael: The Rema is the mara d'asra (the local halachic authority) outside of Eretz Yisrael, and there many communities do wear tefillin on chol hamoed. There are numerous practices in this, such as putting them on at home and removing them before arriving at the shul.
Eretz Yisrael: In Eretz Yisrael the view of Rabbi Yosef Karo was accepted as the mara d'asra, and tefillin are not worn on chol hamoed. There are those in Eretz Yisrael who wear them privately, because this interruption does not sit well with them, but the great majority do not wear them.
The book by Rav Binyamin Shlomo Hamburger: From Bnei Brak, who devotes himself to the customs of Frankfurt and even founded a community using the Frankfurt melodies, a great expert, he published a book of some six hundred pages in which he proves, to his mind conclusively, that one should wear tefillin on chol hamoed even in Eretz Yisrael, and he includes responsa from all the early authorities.
Why the discussion never closes: Every so often a responsum of some early sage surfaces from manuscript, and the picture changes. Even regarding the claim that in Eretz Yisrael they did wear them, a responsum was found stating that in Eretz Yisrael they did not.

(A Midrash is cited that there are 310 days in the year on which tefillin are worn, and from the calculation of the days of the year minus fifty-four Shabbosos, some deduce that tefillin are worn on chol hamoed as well. The exact wording of the Midrash is not recalled precisely.)

Traditions and incidents that have been heard: A tradition that outside of Eretz Yisrael one who worked on chol hamoed wore tefillin and one who did not work did not wear them; and a tradition that in Breslov they did wear them. It was also related about a man from Eretz Yisrael who traveled to Belgium for chol hamoed and davened in a Pshevorsk chassidus community, and since he does not wear tefillin in Eretz Yisrael but did not want to separate himself from the congregation, he asked a dealer in Bnei Brak for tefillin batim from which the parshiyos had been removed, and he wore the empty batim without a berachah.

In passing · The Torah reading on chol hamoed and the recitation of Kaddish

Four aliyos and two scrolls: On chol hamoed four people are called up to the Torah. Three read from the first scroll, and the fourth reads from the second scroll. This is not a maftir, and therefore no Kaddish is said between the reading from the first scroll and the reading from the second.
The comparison to Shabbos: On a Shabbos when two scrolls are taken out, Kaddish is said after the first scroll, because with it the Torah reading of the parshah has been completed, and afterward comes maftir, with the Kaddish separating between them.
The difference: At Shacharis of chol hamoed there is no maftir, and the Torah reading includes both scrolls together. Therefore there is no Kaddish in the middle, and it is said at the conclusion of the reading.

In passing · Transmitting stories and incidents about rabbanim

The pamphlet "Pesi Ya'amin L'chol Davar": A substantial pamphlet published by an erudite talmid chacham in the United States, in which he warns that one must not believe everything told in the name of tzaddikim and chassidim, and that even one who saw something with his own eyes does not necessarily understand what he saw. There was written give and take about the pamphlet, and among those who responded was Rav Yehoshua Mondshine, one of the great experts in the teachings of Chabad, who wrote and reissued the Chabad Sefer HaMinhagim.
An incident with Rav Moshe Feinstein: In his responsa he permitted milk that is under government supervision, and nevertheless he was seen at a wedding lifting a cup of milk and putting it down without drinking. When he was asked whether he had retracted, he answered that he had not retracted at all, but rather that he had lifted the cup, saw that it was empty, and therefore put it down.
An incident with the Chazon Ish: During the siege of Jerusalem a dispute arose in the yeshiva regarding powdered milk of non-Jews, whether the status of chalav akum applies to it after it has undergone processing and water is added to it. One student said the Chazon Ish had permitted it to him and the other said the Chazon Ish had forbidden it to him. When a person of standing was sent to clarify the matter, the Chazon Ish answered that he saw that the first one was in danger, thin and gaunt, and in days of famine in Jerusalem he permitted it to him, whereas the second was healthy and strong and could manage without it.
The conclusion: A person cannot know what lies behind a ruling if he does not know the facts precisely, and therefore stories are transmitted with limited reliability and require checking. Nevertheless, the very act of recording the deeds of tzaddikim is an important thing.