Karpas and why it was chosen

Shulchan Aruch, Siman 473, se'if 4

"They bring before the head of the household a plate that has on it three matzos, maror, charoses and karpas."
(Shulchan Aruch)

Karpas and why it was chosen

"Karpas is a type of vegetable that is called karpas, and they chose this species from the outset because it is an acronym: samech (sixty) and perech, that is to say, six hundred thousand performed backbreaking labor."
(Mishnah Berurah, s.k. 19)

Why a vegetable is needed: For the first dipping we want something whose berachah is borei pri ha'adamah, so that this berachah will also cover the maror that is eaten later on in the seudah.
Why karpas specifically: Many vegetables have the berachah of borei pri ha'adamah, and karpas was chosen because of the hint contained in its name: the letter samech (six hundred thousand) followed by "perech," recalling the backbreaking labor in Mitzrayim.

The names of the foods as the reason they were chosen

Chazeres (lettuce): Of the five species valid for maror, lettuce is given precedence, even though it is not the most bitter of them. The Gemara expounds: chasa, that the Merciful One had pity on us, that the Ribbono shel Olam had compassion and mercy upon us.
Korban Pesach: Even the Torah gives the korban a name derived from the event, because Hashem passed over the homes of Bnei Yisrael.
Hints in names generally: On the night of the Seder there are foods whose very name is what justifies choosing them over other foods, and outside Eretz Yisrael the practice was extended to additional foods, based on the name or on the gematria. This somewhat resembles the custom of the simanim on Rosh Hashanah.
The same in reverse: Some avoid foods on Pesach whose name recalls something forbidden. For that reason there are those who do not eat chummus, because the word resembles "chametz," and likewise there are those who avoid vinegar. This practice exists even among Sephardim, and it has nothing to do with kitniyos.

Which vegetable to take for the first dipping

"Or another vegetable - it is preferable from the outset that he take from that species over which one recites the same berachah that is recited on the maror, so that with this he also covers the maror that he eats during the seudah; but he should not take for this one of the five species of maror."
(Mishnah Berurah, s.k. 20)

Why not one of the maror species: If he takes for the first dipping the very species he is using for maror, it will not look like maror, since he already ate it at the beginning of the Seder.
In practice: The custom for the first dipping is potato, radish, carrot, onion and the like, and not lettuce when the lettuce is being used for maror.
One who uses chrein for maror: Someone who takes horseradish for maror may take karpas for the first dipping, and likewise celery or any vegetable whose berachah is borei pri ha'adamah.

Preparing salt water before Shabbos

Making salt water on Shabbos: It is forbidden to make salt water on Shabbos, so one should prepare it before Shabbos and not on Shabbos itself.
On Yom Tov: Some hold that it is forbidden to make salt water even on Yom Tov, and according to this view the salt water should already be prepared on Friday for the night of the Seder.
If he forgot: One who forgot may make a small amount of salt water, not a large quantity, and some say he should do it with a change in the normal manner: he should pour the water onto the salt rather than the salt into the water.
For the sake of a mitzvah: Since the dipping is for the sake of a mitzvah, there is room to permit making it on Yom Tov, but it is preferable to prepare it from Friday.

The reason for the first dipping: to arouse the children's questions

"Now, as for the dipping of the vegetable in liquid, this is a takanah of the Chachamim, in order to astonish the children, who will see the change, that vegetables are eaten with a dipping, which is not the way they are eaten before the seudah all year round, and they will ask about this change."
(Mishnah Berurah)

The dipping as a change: Eating a dipped vegetable at the start of the seudah is not the normal practice during the year, and this is what prompts the child to ask.
In "Mah Nishtanah": This very change is mentioned among the questions: "On all other nights we do not dip even once, and on this night twice."
The Haggadah as an answer: The mitzvah of saying the Haggadah is that it be said as an answer to questions that were asked, as the pasuk says, "When your son asks you," and the answer begins, "We were slaves." The son asks and the father answers, and through this "you shall tell your son" is fulfilled.

Learning through question and answer

The value of a question: When a child asks, he becomes a participant in the matter, and therefore he remembers more. In the view of Chazal this is the ideal way to teach.
A hint from the order of the Mishnah: The very first Mishnah does not open by stating the halachah but with a question, "From when does one recite Shema in the evening?", and only afterward, "From the time the Kohanim enter to eat their terumah." Here too the teaching is by way of question and answer.
The four sons: In the Haggadah as well, "the wise son, what does he say?", and the wise son opens with a question.

The two cooked foods: a remembrance of the Pesach and of the chagigah

The korbanos of one who comes up for the festival

Shalmei simchah: One who comes up for the festival is obligated to rejoice on the festival, and there is no rejoicing without meat and wine, so he brings a korban of shalmei simchah, slaughtering it on the eve of the festival and eating it on the festival.
Olas re'iyah: One who comes to be seen in the Beis HaMikdash brings an olah, which is entirely consumed for Hashem.
Shalmei chagigah: The festival chagigah offering. It emerges that one who comes up for the festival brings three korbanos with him, and this is no simple matter: he comes up with his entire family and with a whole flock.

Pesach and Chag HaMatzos: two separate matters

Do not confuse them: Chag HaPesach is the fourteenth of Nisan, and Chag HaMatzos is from the fifteenth of Nisan onward.
The mitzvah of coming up for the festival: The mitzvah of aliyah la'regel does not apply to the Pesach but to Chag HaMatzos, and on Chag HaMatzos a person is obligated in all three korbanos: shalmei simchah, olas re'iyah and shalmei chagigah, just as on the other festivals.
Coming up on the fourteenth of Nisan: On the fourteenth of Nisan a person goes up to Yerushalayim because he must slaughter the Korban Pesach there, not because of the mitzvah of re'iyah. One who is not bringing a Korban Pesach, for whatever reason, is not required to come up at this stage; his obligation is on Chag HaMatzos.
The order of the Pesach in the Mikdash: The korban is slaughtered on the fourteenth in the afternoon, roasted, and eaten at night in a group, and from here comes the family character of the night.

Chagigas arba'ah asar

The laws of the Korban Pesach: It is eaten only at night, in a group, and only until chatzos according to how we rule (there is an opinion that it may be eaten until dawn). For this reason one must finish eating the afikoman before chatzos, for otherwise it becomes nosar. It is also said of it, "and you shall not break a bone in it."
The Pesach is eaten al hasova: It is not eaten ravenously, but as a settled, dignified person eats, neither out of hunger nor as one who is already completely full.
The concern: A person who has come up to Yerushalayim with his family, and who on erev Pesach cannot eat until the korban is slaughtered and roasted at night, will end up eating with eagerness and hunger, and out of that he might even break a bone; and that is not the manner of freedom, since one must eat the way kings eat.
The takanah: The Chachamim found a hint in the Torah and instituted that before eating the Korban Pesach a person should eat from a chagigah offering that comes together with the Pesach, called chagigas arba'ah asar. This is not the festival chagigah mentioned above, but a chagigah connected to the Korban Pesach, so that he will eat the Pesach calmly and al hasova.

Meat and egg on the plate

"And they take two cooked foods, one as a remembrance of the Pesach and one as a remembrance of the chagigah... and the custom is meat and an egg."
(Shulchan Aruch)

What the two cooked foods recall: One is a remembrance of the Korban Pesach and one a remembrance of the chagigas arba'ah asar that was eaten together with it, for the night of the Seder is a remembrance of what our forefathers did in the Mikdash.
In the time of the Gemara: The custom was two kinds of meat, one for the Pesach and one for the chagigah, and over time the practice became meat and an egg.

"And the custom is meat and an egg - and all the more so, if one wishes to make two kinds of meat, one roasted and one cooked, as a remembrance of the Pesach and the chagigah, that is perfectly fine; it is only that with an egg too one fulfills this."
(Mishnah Berurah, s.k. 23)

Roasted and cooked: The Pesach is eaten only roasted, whereas the chagigah may even be cooked, so one who takes two kinds of meat takes one roasted and one cooked.
Why the custom became an egg: So that a person should not become confused as to which meat represents the Pesach and which the chagigah, they distinguished between them by taking meat and an egg.

"And some have written that the reason they chose bei'a is because of: the Merciful One wished to redeem us with an outstretched arm."
(Mishnah Berurah)

The hint in the name: "Bei'a" in Aramaic is an egg, and "ba'a" means "wanted": the Merciful One wanted to redeem us with an outstretched arm. That is why it is specifically a zeroa (arm) and an egg that are placed on the plate. Once again we find that the name of the food is the reason for choosing it.
An additional reason: Some have written that we are making a remembrance of mourning over the Beis HaMikdash, since we cannot bring a Korban Pesach, and an egg is the food of mourners, because it is round like the wheel that turns in the world.

The law of the roasted zeroa on the night of the Seder

It is not eaten at night: It is forbidden to eat the roasted zeroa on the night of the Seder, because it looks as though one is eating kodshim outside the Mikdash, since there is no Beis HaMikdash; and since it is a remembrance of the Korban Pesach, people will say that he is actually eating a Korban Pesach.
In practice: He should eat it the next day, during the day, and not at night.

Erev Pesach that falls on Shabbos

"A remembrance of the chagigah - and if erev Pesach falls on Shabbos, some say that this is not needed, and it is enough that there be one corresponding to the Pesach, for the chagigah does not come then, since it does not override Shabbos... and some say that nevertheless one must make two cooked foods just as in other years, for the Rabbanan did not draw distinctions in this, since it is only a remembrance."
(Mishnah Berurah, s.k. 24)

The case: In a year when erev Pesach falls on Shabbos, the night of the Seder is on Motzaei Shabbos, and if the Beis HaMikdash were standing the Korban Pesach would be slaughtered on Shabbos itself.
Which korban overrides Shabbos: A communal korban overrides Shabbos, such as the two temidim and the musaf of the festival, but an individual's korban does not override Shabbos. The festival chagigah that a person brings for himself is not slaughtered on Shabbos, and even regarding Yom Tov the matter is discussed.
The status of the Korban Pesach: Even though it is the korban of a group, it has laws that allow it to be slaughtered on Shabbos. The slaughtering and the removal of the emurim override Shabbos, but the roasting does not override Shabbos, and therefore it is slaughtered on Shabbos afternoon and roasted at night.
Hence there is no chagigah: The chagigas arba'ah asar is a full-fledged individual korban, and it cannot be brought on Shabbos. It emerges that in such a year there was no chagigah at all.
The practical difference on the plate: According to one view, since there is no chagigah there is no need for a second cooked food, and one corresponding to the Pesach suffices, so seemingly no egg is needed. According to the other view, one makes two cooked foods just as in other years, since the Chachamim drew no distinction in this, given that it is only a remembrance.
In practice: The custom is to have both cooked foods, and there is no special plate for a Seder night that falls on Motzaei Shabbos.

(Even in the time of the Beis HaMikdash this was not a simple matter: this halachah was forgotten, since erev Pesach falls on Shabbos sometimes after only a few years and sometimes only after twenty years, and the Bnei Beseira did not know whether the Pesach overrides Shabbos or not.)

In passing · Avoiding vinegar on Pesach

The custom of avoiding it: There are those who avoid vinegar on Pesach, and in general there are those who will not touch sour foods.
The issue is not the sourness: There is no problem with lemon on Pesach, even though it is very sour; the concern with vinegar is because of an additional ingredient mixed into it.