Gemara

(Abaye): It is a Mitzvah to count the days, for it says "Tisperu Chamishim Yom". It is also a Mitzvah to count the weeks, for it says "Shiv'ah Shavu'os Tispar Lach", and it is called "Chag ha'Shavu'os".

Menachos 65b (Beraisa): "U'Sfartem Lachem" - it is a Mitzvah for every individual to count.

66a: Rabanan of Rav Ashi's academy counted the days and the weeks.

Ameimar counted the days but not the weeks;

He holds that there is no Mitzvah today, only a commemoration of the Mitzvah when the Mikdash stands. Therefore, it suffices to count the days.

Rishonim

Rif (Pesachim 28a): Just like it is a Mitzvah to count the days, it is also a Mitzvah to count the weeks, like Abaye taught.

Ran (DH u'Mitzvah): We hold like Abaye and the Rabanan of Rav Ashi's academy against Ameimar. Most Meforshim hold that nowadays that we do not bring Korbanos, Sefiras ha'Omer is only mid'Rabanan.

Kaf ha'Chayim (5): This is not a contradiction, for Abaye and Rabanan of Rav Ashi's academy could agree that Sefiras ha'Omer is mid'Rabanan today.

Rambam (Hilchos Temidim 7:22): It is a Mitzvas Aseh to count seven full weeks from the day the Omer is brought - "U'Sfartem Lachem mi'Macharas ha'Shabbos... Sheva Shabbosos". It is a Mitzvah to count the days with the weeks - "Tisperu Chamishim Yom". We count at the beginning of the day. Therefore, we count from the night of the 16th of Nisan.

Ran (DH uv'Hagadah): It seems that the Rambam holds that those who require counting the days and weeks hold that Sefiras ha'Omer is mid'Oraisa even nowadays.

Question: A Zavah must count seven clean days, but she does not bless and she need not count out loud. Why is Sefiras ha'Omer different?

Answer #1 (brought in Radvaz): Chachamim did not enact that she bless because if she will see blood in the middle, this cancels what she counted. Since she need not bless, she need not count out loud.

Answer #2 (Radvaz): Chachamim obligate to bless only on obligatory counts. A Zavah may opt not to count and to remain Teme'ah, and never have relations, enter the Mikdash or eat Kodesh.

Note: This is like the opinion that Korban Pesach is optional for women. If she is married, she would need her husband's consent to refrain from relations, e.g. if he already fulfilled Peru u'Rvu. If the Radvaz means that for some it is not obligatory, e.g. for single women, we can also answer why no Berachah was enacted for Zavim, for some men are exempt from Korban Pesach. Tosfos (Pesachim 3b DH me'Alyah) exempts one who cannot walk, does not own land in Eretz Yisrael or lives in Chutz la'Aretz.

Ri Korkus and Ran (ibid.): Some explain that the Mitzvah to count weeks is only on days 7, 14,..., which complete a number of full weeks. Nowadays we are stringent to mention the number of weeks and days on every night. The Rambam says 'it is a Mitzvah to count the days with the weeks', as opposed to 'and the weeks'. This connotes that he agrees with our custom.

Rosh (Pesachim 10:40): Just like it is a Mitzvah to count the days, it is also a Mitzvah to count the weeks. Rabanan of Rav Ashi's academy counted the days and the weeks. Ameimar counted the days but not the weeks, for he holds that it is only a commemoration of the Mikdash. The Ri permits to bless when it is Safek night, for it is a Safek mid'Rabanan (nowadays). It seems that it is better to bless shortly before dark, to fulfill "Temimos" (full weeks).

Rebuttal (Ran DH uv'Hagadah): One should not enter a Safek (Berachah l'Vatalah). We need not be more stringent about Temimos nowadays that Sefiras ha'Omer is mid'Rabanan than when it was mid'Oraisa! As long as one counts at night, this is Temimos.

Poskim

Shulchan Aruch (OC 489:1): On the second night of Pesach after Ma'ariv we begin to count the Omer. If one forgot to count at the beginning of the night, he may count at any time during the night. It is a Mitzvah for everyone to count for himself, standing, and to bless beforehand.

Taz (1): We read "Ba'Kamah Tachel Lispor" like 'b'Komah', i.e. standing.

Magen Avraham (2): One may be Yotzei the Berachah from the Shali'ach Tzibur. However, the Beis Yosef (DH Kosav ha'Rashba) says that one may be Yotzei even counting. This is difficult, for "U'Sfartem Lachem" obligates every individual to count! Perhaps it is as if they counted, because Shome'a k'Oneh.

Mishnah Berurah (5): Acharonim argue about whether or not Shome'a k'Oneh applies to counting, but all agree that it applies to the Berachah. However, it is the custom for everyone to count and bless by himself.

Magen Avraham (2): One who counted in Lashon ha'Kodesh without understanding the words was not Yotzei. One may count in any language he understands.

Kaf ha'Chayim (20): She'alas Ya'avetz (139) says that one is Yotzei in Lashon ha'Kodesh even without understanding. Since Poskim disagree, one should count again without a Berachah.

Magen Avraham (1): Women are exempt because the Mitzvah is Zman Grama, but they accepted it on themselves like an obligation.

Mishnah Berurah (3): In our lands women do not count. The Shulchan Shel Shlomo says that in any case they should not bless, for they will surely err one day. Also, in most cases they do not know the meaning of the words.

Note: The Shulchan Shel Shlomo's first concern is lest they continue to count with a Berachah after missing a day. He is not concerned that the Berachos before are retroactively l'Vatalah. Even though one may count in any language, perhaps women will want to count in Lashon ha'Kodesh, even though they do not understand it.

Shulchan Aruch (ibid.): He counts the days and the weeks... Upon reaching seven days he says 'today is seven days, which is one week (ba'Omer)', 'today is eight days, which is one week and one day (ba'Omer)'...

Mishnah Berurah (7): If one counted days but not weeks, he should count again without a Berachah. Even if he did not count again, he may count future nights with a Berachah.

Mishnah Berurah (8,9): Even if one said 'today is such and such day' without saying 'in the Omer', he was Yotzei. L'Chatchilah, we say the units before the tens (e.g. three and 20 days). Matters like this are not Me'akev.

Shulchan Aruch (2): If one counted while it is still day, he must count again. Meticulous people do not count before Tzeis ha'Kochavim; it is proper to do so.

Source (Beis Yosef DH uvi'Seshuvas): The Rashba says so (Teshuvah 1:458). This is the custom everywhere.

Mishnah Berurah (14): Letter of the law, one may rely on the majority of Poskim to count Bein ha'Shemashos, which Safek night, because most say that nowadays Sefirah is mid'Rabanan. It is best to avoid the Safek. If one counted Bein ha'Shemashos he was Yotzei. The Eliyahu Rabah says that he should count again at night without a Berachah.

Kaf ha'Chayim (6,12): Mekubalim say that one should count the first night after the Seder (in Chutz la'Aretz). This has a basis Al Pi Sod, but one should not do so, lest he forget. Further, one who delays counting on any night is seized by Chitzonim.

Shulchan Aruch (3): If one prays Ma'ariv with a Tzibur before night, he should count with them without a Berachah. If he remembers at night, he counts again with a Berachah.

Source (Beis Yosef DH Kosav ha'Rav citing the Avudraham): Machzor Vitri says to do so. If he forgets to count at night, at least he counted the days and weeks. However, the Rambam says that one who counted without a Berachah was Yotzei. This is the primary opinion.

Question (Taz 6 and Magen Avraham 7): Why does the Shulchan Aruch bring the Avudraham's counsel? The Beis Yosef says that the Avudraham retracted and agreed with the Rambam!

Answer #1 (Taz 6): The Shulchan Aruch does not discuss a Tzibur of sinners who count and bless during the day. Rather, it is Bein ha'Shemashos. One who prefers to count after Tzeis ha'Kochavim should count with them lest it look like he does not want to count.

Answer #2 (Magen Avraham 7): The Shulchan Aruch (60:4) holds that Mitzvos require intent. Therefore, one may stipulate not to be Yotzei now if he will remember later.

Magen Avraham (7): One may count before Ma'ariv. On Motza'ei Shabbos one may count before Havdalah. Chanukas ha'Bayis questions this, for it is still Shabbos. Really, it is not Shabbos, just one may not do Melachah until Havdalah.

Rema: Even if he answered Amen to the Berachah of the Tzibur, if he intended not to be Yotzei he counts again with a Berachah.

Magen Avraham (8): Since Mitzvos require intent, even Stam (if he had no intent) he counts again with a Berachah. The Rema suggests negative intent because some say that Mitzvos do not require intent.

Shulchan Aruch (4): If one is asked what day it is, he should not give tonight's count before counting, for if so he may not bless afterwards. Rather, he should say what yesterday was. Before Bein ha'Shemashos we are not concerned for this, for it is not yet time to count.

Mishnah Berurah (22): If this occurred after six days, even if he said tonight's count he may bless, since he did not mention the weeks. (This proves that he did not intend to be Yotzei now.)