Gemara

41b (R. Yochanan): One may bless on the new moon until its deficiency is filled.

Question: When is this?

Answer #1 (R. Yakov bar Idi): It is until seven days after the Molad (new moon).

Answer #2 (Chachamim of Neharde'a): It is until the 16th of the month.

Both hold like R. Yochanan. R. Yakov bar Idi understands that it is until the deficiency of the crescent is filled (i.e. half the circle is visible). Neharda'ei understand that it is until the entire circle is filled.

Rishonim

The Rif (Sanhedrin 14a and Berachos 21a) brings our Gemara.

Talmidei R. Yonah (DH Neharda'ei): Our Gemara discusses the latest time to bless. Maseches Sofrim (20) says that the earliest time is from when 'Tisbasem'. Some explain that this is Motza'ei Shabbos, when we bless on Besamim (fragrances). This is unreasonable. If Rosh Chodesh is on Sunday, why should we wait so long?! Some say that it is when the moon gets a bit big and illuminates. R. Yonah says that 'Tisbasem' refers to sweetness, after after two or three days, when one can benefit from its light.

Rambam (Hilchos Berachos 10:17): If one did not bless on the first night, he blesses until the 16th of the month, until the deficiency is filled.

Rosh (5:1): The Halachah follows Neharda'ei, for they are the majority.

Poskim

Shulchan Aruch (OC 426:3): We bless on the moon until 16 days from the day of the Molad, not including day 16.

Beis Yosef (DH Omar R. Yochanan): Kol Bo says that l'Chatchilah one blesses within seven days. If one delayed, he blesses until 16 days, but no later. This is because the sun is opposite the moon on the 15th of the month, so the radiance of the moon is strong until it is full. When the sun is in the west and the moon is in the east, the distance between them is great. We bless until the deficiency is filled, for after mid-month the moon gets closer to the sun and its light dims. We bless on increase, which is good, but not on decrease.

Rema: One should be Mekadesh only until half of 29 days, 12 hours and 793 parts (out of 1080) of an hour from the Molad.

Magen Avraham (12): The Mechaber connotes that we count from the day of the Molad, and not from hour to hour. E.g. if the Molad was four hours before dark on Sunday afternoon, one may not be Mekadesh (15 days later) on Sunday night. The Rema allows 14:18:396.5 (14 days, 18 hours and 396.5 parts) from the Molad. If so, he allows more time. His words ('only until') connote that he allows less time! The Tur allows 15 days from the time of the Molad. If the Molad was two or three hours into the night on Motza'ei Shabbos, one may be Mekadesh it at the beginning of Sunday night 16 days later. The Rema forbids.

Beis Yosef (DH u'Mah): The Maharal (19) says that we count the 16 days from the Molad, for it depends on filling the deficiency. It is not really 16 or 15, rather, the majority of 29:12:793. I say that if there was a lunar eclipse (the moon is in the earth's shadow), the moon was exactly opposite the sun, so one may not bless after this. Stam (when there is no eclipse), we rely on the calculation we received from our Rebbeyim, which is the average interval between one Molad and the next. This is even if there was a solar eclipse (the moon blocks the sun's light), which is the true Molad, before or after our calculation. The Rambam wrote that we push off Rosh Hashanah (from falling on Sunday, Wednesday or Friday) to correspond to the true Molad, which differs from the calculation.

Darchei Moshe (1): I do not understand. The Rambam says that we push off Rosh Hashanah (from certain days), for usually this results in the true Molad, but he explicitly says that sometimes the true Molad is before or after! Also, the Mefaresh in Hilchos Kidush ha'Chodesh brought proofs to refute the Rambam. Perhaps we can reolve the Maharil. When there is lunar eclipse, we see that it is truly in the middle, for it is a Shinuy in the entire world (all who can see it, i.e. those for whom it is night, see it at the same time). A solar eclipse changes based on the place and the people; one who understands eclipses know this. (A tiny part of the earth is darkened at any moment. The place of the eclipse traverses the earth in at most four hours - PF.) The Maharil brought the Rambam only to prove that usually, our Molad is correct. Therefore, one relies on it unless he knows that the true Molad is earlier, and we can know this only through a lunar eclipse.

(Hagahos Tur ha'Shalem 14): Divrei Torah (6:14) says that the Beis Yosef retracted, therefore, he did not bring this in Shulchan Aruch.

Bi'ur Halachah (DH v'Lo): Some (including the Ramah) allow blessing until 16 days. If one did not bless before, some say to bless on night 16 without Hash-m's name. On night 15, if it is after 14:18:396.5, perhaps one may rely on those who permit to bless with Hash-m's name.

Kaf ha'Chayim (53): One should not bless (with Hash-m's name) after 14:18:396.5. We do not bless when in Safek, even against the Mechaber, even if the majority say to bless.

Kaf ha'Chayim (59): If one needs to say Kri'as Shema and its Berachos, and also Kidush Levanah, all agree that Kri'as Shema has precedence, for it occurs more regularly and it is mid'Oraisa. However, if it is near the last time to be Mekadesh it and there is concern lest it be covered with clouds, one should say Kidush Levanah first.

Tiferes Yisrael (Shvilei ha'Raki'a (before Seder Mo'ed) 2,3): The Molad is the earliest time that a sliver of the moon is lit after it was hidden. This varies according to one's location on earth. Since the Sanhedrin used to be Mekadesh in Yerushalayim, to commemorate the Mikdash our custom is to rely on the calculation of when the Molad was in Yerushalayim.

Rebuttal (Mishneh Hilchos 11:344): Almost all Poskim hold that the Molad is when the moon is under the sun (i.e. directly below or above the line connecting the centers of the earth and the sun - PF). It is the same for everyone. Many Luchos give the time of the Molad in Yerushalayim. One in a different time zone must fix this (to know the times for Kidush Levanah).

Luach Itim l'Binah (b'Sof p.5, Zman Birkas ha'Levanah 1): The custom is to announce at Kidush ha'Chodesh the calculated Molad, even though it is not the true Molad. Opinions differ about how to adapt our tradition for the calculation to today's clock system that begins each day at 12:00. (Note: The interval between one Molad and the next varies based on the distance of the earth from the sun; the earth is closer in winter. The true Molad is sometimes 15 hours different than the calculated time - PF.)

Shulchan Aruch (4): We do not bless on it until seven days have passed.

Magen Avraham (13): The Rambam says that if one did not bless on the first night, he blesses until the 16th. The Maharshal inferred that one need not wait three days. Sefer ha'Kanah says that the ideal Mitzvah is to be Mekadesh it immediately. Hagahos Maimoniyos says that one should wait seven days. The Levush and Bach say that if Motza'ei Shabbos is less than seven days from the Molad, we rely on R. Yonah and Mekadesh it if it is after three, for it is Motza'ei Shabbos and the Tzibur is gathered, and "b'Rav Am Hadras Melech" (it is best to do Mitzvos with many others). We are not Mekadesh within three days, for one can wait until the next Motza'ei Shabbos.

Mishnah Berurah (20): Most Acharonim disagree. In any case, 72 hours after the Molad, one can benefit from the light, so he should not delay the Mitzvah. If this is in the middle of the week, he should wait until Motza'ei Shabbos. Several Acharonim including the Gra hold that one should never delay. One who does so certainly has whom to rely on, especially in winter (when there might be several cloudy nights in a row and there is danger of losing the Mitzvah) it is praiseworthy to be Mekadesh promptly.

Kaf ha'Chayim (61,62): Based on Kabalah, one should not be Mekadesh before seven full days.