Siman 489, Se'if 1

Shulchan Aruch, Siman 489, se'if 1
"On the second night, after Maariv, we begin counting Sefiras HaOmer. If one forgot to count at the beginning of the night, he may go and count all night long. And it is a mitzvah for each person to count for himself."
(Shulchan Aruch)

Where the Counting Fits Into Maariv

The piyutim: In se'if katan 1 the Mishnah Berurah discusses the piyutim of the second night. We do not have the practice of saying them, so this is not our concern here.

"After Maariv, before Aleinu; and we push it as early as we possibly can, so that 'they shall be complete' is better fulfilled."
(Mishnah Berurah, s.k. 2)

What "they shall be complete" means: The verse says, "seven complete weeks shall there be." From the words of the Mishnah Berurah it emerges that the completeness does not refer only to the weeks, that they be full weeks, but to each individual day as well.
The practical upshot: When a person says "today is one day of the Omer," "today is two days of the Omer," it should ideally be a full day. That is why we try to count immediately at the transition from day to night, when the new day begins at tzeis hakochavim (nightfall).
How early can we go: One cannot count before Krias Shema and Shemoneh Esrei, but before Aleinu is possible, and therefore according to this view we count before Aleinu.

The View of the Mor uKetziah: Counting at the End of Davening

(The Mor uKetziah is a commentary on the Shulchan Aruch by the son of the Chacham Tzvi, Rav Yaakov Emden.)

"Nowadays we count after the davening... because in earlier generations the practice was to daven Maariv before nightfall."
(Biur Halachah, citing the Mor uKetziah)

The background in earlier generations: It was hard to gather a minyan of working men; if they went home after Minchah they would not come back for Maariv. So Minchah and Maariv were joined together, and the result was that Maariv was davened early.
What that meant for the counting: Since Maariv was moved up, night had barely begun by the time davening ended. Hence the practice arose to count at the end of davening, after Aleinu, in order to be certain that the time for counting had arrived.
Someone who davens late: According to this approach, a person who davens Maariv at nine o'clock at night, where there is no question at all that the time has come, should count before Aleinu.

The View of the Chok Yaakov: The More Frequent Comes First

"For by strict law one must give precedence to Krias Shema and the tefillah, which are the more frequent."
(Chok Yaakov, cited in the Mishnah Berurah)

The reasoning: Krias Shema and Shemoneh Esrei are said every single day, while Sefiras HaOmer applies only during a certain period of the year. Where one mitzvah is frequent and one is not, the frequent one comes first. Therefore the davening precedes the counting, and we do not push the counting off until after Aleinu.
The two customs: From here the minhagim diverge to this very day, whether one counts before Aleinu or after Aleinu.

Explaining the Dispute According to Rav Yosef Meir Soloveitchik (the Rav of Boston)

Rav Soloveitchik traces the dispute to a fundamental question: what is the status of Aleinu? Is it part of the davening, or is it a prayer said upon leaving the shul?

The view of the Bach: Aleinu is said when one leaves the shul. Inside the shul a person is immersed in an atmosphere of kedushah and prayer; when he steps outside he encounters idolaters and all sorts of characters. So before he goes out he strengthens himself and says, "He has not made our portion like theirs," for they bow to emptiness and nothingness.
The implication according to the Bach: Aleinu is not an organic part of the davening but a prayer of departure. The davening itself concludes with Uva LeTzion, Kaddish and Shir shel Yom, and therefore it makes sense to count before Aleinu. Along these lines, someone who stays in shul to learn Daf Yomi could arguably say Aleinu at the point when he leaves.
The view of the Arizal: The Arizal established that one must say Aleinu after each and every tefillah, which implies that it is an inseparable part of the davening.
The implication according to the Arizal: If Aleinu is part of the davening, the counting must come after it, since the counting cannot precede a part of the davening.

Proofs and Practices Regarding Aleinu

Shacharis on a day with Musaf: We do not say Aleinu at the end of Shacharis but only at the end of Musaf, even though Shacharis has been completed. This shows that Aleinu is not the conclusion of the davening but the point of departure from shul.
Minchah on Yom Kippur: We do not say Aleinu at the end of Minchah on Yom Kippur, whereas according to the reasoning that Aleinu is said at the end of every tefillah it should have been said. The explanation is that in earlier times people did not leave shul on Yom Kippur; they remained there the entire day.
Joining with the congregation: The Acharonim ruled that someone who has finished his own davening and is in shul when the congregation says Aleinu should say it along with them, because it is an acceptance of the sovereignty of Heaven. Similarly, someone who is in shul when the Thirteen Attributes of Mercy are being said joins with the congregation. This demonstrates the importance of this prayer.
The origin of the prayer: According to many, the source of Aleinu LeShabeiach is the Rosh Hashanah davening, in the Malchuyos section of Musaf, and because of its importance it was transferred to the tefillos of the whole year.
Aleinu at the end of a bris milah: Although it seems to have no connection to a bris, the fact that it is said there shows the importance of the prayer, and also fits its content: "He has not made our portion like theirs." We praise the Ribbono shel Olam for having chosen us, and here this child is now joining Klal Yisrael.
Aleinu at Kiddush Levanah: The Mishnah Berurah explains that we say Aleinu after Kiddush Levanah so that people should not say he is bowing to the moon; he therefore announces that the praise belongs to the Ribbono shel Olam alone.

Women and Slaves and Sefiras HaOmer

"Women and slaves are exempt from this mitzvah, since it is a positive mitzvah that is time-bound."
(Mishnah Berurah, s.k. 3)

Which slaves: Wherever "slaves" is stated without qualification, it does not refer to a Jewish servant, who is a Jew in every respect and obligated in all the mitzvos, but to a Canaanite slave. Such a person has accepted a partial obligation in mitzvos and his status is like that of a woman, which is why he is listed together with women in the same category.
The Magen Avraham: "However, they have already taken it upon themselves as an obligation." That is, Jewish women accepted the counting upon themselves as an obligation, and there are those who in fact count.
The Mishnah Berurah: In our times women do not have the practice of counting at all, and the custom is not uniform.
The Shulchan Shlomo: In any event they should not recite the berachah, his reason being that they do not know the meaning of the words and the thing is not done properly. This reason was given in light of the situation in his generation.
In practice: There is no prohibition in the counting itself, and a woman who wishes to count may count. One who wishes to recite the berachah, like women who make a berachah on lulav, needs to realize that she is taking upon herself all forty-nine days, not counting today and not counting tomorrow.

Why We Do Not Recite Shehecheyanu on Sefiras HaOmer

Sefiras HaOmer is a mitzvah that comes once a year, and the Baal HaMaor asks why we do not recite Shehecheyanu at the first counting.

Rejecting the Claim That the Mitzvah May Be Interrupted

The claim: Some want to say that we do not recite it because there is no guarantee that a person will complete all forty-nine days, since he may forget in the middle.
The Rashba's answer in a parallel case: The Rashba was asked about a Jew who washed his hands and recited Al Netilas Yadayim, and before he ate the cat ate the bread and he had no bread left. Must he say "Baruch shem kevod malchuso le'olam va'ed" because of a berachah levatalah? The Rashba answered that he need not, since he performed the mitzvah properly, washed and recited the berachah, and the fact that the bread disappeared afterwards is not his fault.
The conclusion for our case: So too with the counting: at the moment he counts for the first time everything is in order, and if afterwards he becomes ill or forgets, that has nothing to do with the berachah he recited at the outset.

The Baal HaMaor's Answer, and the Comparison to Bedikas Chametz

The parallel to bedikas chametz: Bedikas chametz is also a mitzvah whose berachah comes once a year, and we do not recite Shehecheyanu over it. One of the reasons offered is that nobody is happy about bedikas chametz: in earlier generations a person would find in his house food he had bought at a bargain and now had to throw it all out, and his heart aches. Nowadays we sell the chametz, and not everyone is entirely satisfied with that arrangement.
The Baal HaMaor's answer regarding Sefiras HaOmer: After the counting we say, "May it be His will that the Beis HaMikdash be rebuilt speedily in our days," for we have no Beis HaMikdash and the Omer offering is not brought. In the very same breath in which a person would be reciting Shehecheyanu he is expressing his grief over the destruction, and therefore we do not say Shehecheyanu over a time in which there is no Beis HaMikdash.

In passing · Versions of the Davening and the Customs of Different Communities

How many customs there are in Aleinu: Aleinu is not simple at all. There are communities that do not say it at all, not on Yom Kippur and not the rest of the year, and there are those that say only half of Aleinu, for example saying "Aleinu LeShabeiach" but not "Al Kein Nekaveh Lach." The Italian custom, for instance, differs in this matter.
A person's habit is not the whole picture: From the moment a person is born, picks up a particular siddur and davens in a particular place, he does not know the other possibilities and assumes his nusach is the only one. In the core parts of the davening everyone is the same, but in the additional prayers there is no uniformity.
The many siddurim: We are not talking only about Ashkenazi and Sefardi. Until a hundred years ago there was no such concept as "Edos HaMizrach" at all, and even that name is a general one that includes Yemenite, Tunisian, Algerian, Moroccan and more, each with its own siddur and its own customs.
How customs changed in Eretz Yisrael: One who researches minhagim cannot rely on what he sees in a shul in Israel, because since the waves of immigration some of the original customs were forgotten and were influenced by the Ashkenazim. One has to go back to the original siddurim and to the practices that were followed in the countries of origin.
A comment of Rav Ovadia Yosef: When he was asked about the stringencies of Pesach he said that we Sefardim came to Eretz Yisrael and were influenced by the Ashkenazim, and worse than that, the Ashkenazim are influenced by "Ashkenaziness."
The Frankfurt minyan: Rav Hamburger in Bnei Brak founded a minyan following the custom of Frankfurt, the custom of the Yekkes, because in Israel the customs had changed, and he wanted to preserve the original practice as it once was.

In passing · Berachos Recited Only Once a Year

Bedikas chametz: Its berachah is recited once a year, both in Eretz Yisrael and outside it.
Birkas HaIlanos (the blessing on blossoming trees): This too is among the berachos recited only once a year.
Other berachos: There are additional berachos in this category, and it is worth looking for them.

In passing · Shehecheyanu at a Bris Milah

Differing customs: Outside of Eretz Yisrael Shehecheyanu is not recited at a bris milah, while in Eretz Yisrael it is. The very existence of two customs shows that there is a genuine discussion here, unlike bedikas chametz, where all agree that no berachah is said.
The argument from the pain: Some wanted to say that we do not recite it because the baby is in pain. But the baby is not the one reciting the berachah; the father recites it, and he is happy to have merited a son.
How the practice was decided in Eretz Yisrael: In Eretz Yisrael we follow Maran Rabbi Yosef Karo, who is the mara d'asra (the halachic authority of the place), and he ruled to recite the berachah, while the position of the Rema there is not clear. Therefore in Eretz Yisrael we recite it and outside of Eretz Yisrael we do not.