Rosh Hashanah, Mishnah 1. The Mishnah opens with two situations in which the court did not manage to sanctify the new month in its proper time.
The two situations in the Mishnah:
"Ra'uhu beis din v'chol Yisrael" - the court itself, together with a large public (not necessarily everyone), saw the new moon.
"Nechkeru ha'eidim" - even though the new moon was not seen by the court or by the public, the witnesses were already examined and found valid and trustworthy.
In both of these cases the court had no opportunity, for whatever reason, to actually and formally declare the day "mekudash" - sanctified as Rosh Chodesh - before nightfall. And the halachah is that the month is not sanctified at night, so once it grew dark, the time was gone.
Therefore, even though everyone saw the new moon, or the witnesses had already been examined and found trustworthy, in practice the day was not formally established as Rosh Chodesh. Rosh Chodesh is pushed off by one day: the month is made full and counts thirty days, and the second day is Rosh Chodesh.
The court alone saw it:
The Mishnah continues and asks: what is the law when the court itself saw the moon, and there were no witnesses at all? The answer is that two of the court stand up and testify before the remaining members of the court, and those members declare and formally sanctify the month on the basis of their testimony, since the members of the court did in fact see it.
A court of three:
The previous case dealt with a court made up of many members, whereas here the entire court consists of only three. Two of those three who saw the new moon stand up and serve as witnesses, and they bring in others of their colleagues who did not see the moon but are fit to serve as judges, and seat them together with the one remaining judge. The two testify before them, and they declare the month sanctified.
The reason for adding two more judges to the one original member of the court who remained is that an individual is not trusted on his own, and he cannot serve as a court by himself. There is no such thing as a court of one man, and at least three judges are required.
When is testimony required and when does seeing suffice?
The last two cases, whether in a large court or in a court of three, deal - as the Gemara explains - with seeing the moon on the night of the thirtieth, at the onset of the thirtieth day, that is, after the twenty ninth day. And since they saw it at night, the month could not be sanctified then, and testimony about seeing the moon is also not accepted at night. Therefore, when the following day arrives, testimony must be given.
The Gemara adds, however: had they seen the new moon before nightfall, at a time when the month could be sanctified, there would be no need for two of them to testify, neither in a large court nor in a court of three. The reason is stated in the Gemara's well known phrase: "Lo tehei shemi'ah gedolah mer'iyah" - hearing testimony is not superior to actually seeing with one's own eyes. Once they saw with their own eyes at a time when the month could be sanctified and at a time when testimony could be accepted, their very seeing is like accepting testimony, and they can sanctify the month on that basis with no testimony at all.
It emerges that only if they saw the new moon at a time when the month is not sanctified and testimony is not accepted are they required to testify the next day; but if they saw it earlier, on the twenty ninth day, they sanctify the thirtieth day as the first day, as the day of Rosh Chodesh.
In summary: in this Mishnah we learned that if the court and all of Israel saw the new moon, or the witnesses were examined and found valid, and they did not manage to say "mekudash" before nightfall, the month is made full and Rosh Chodesh is pushed off to the second day. If only the court saw it, two of them stand up and testify before the remaining judges. If the court consisted of three, two testify, and other colleagues are brought in as judges to join the one who remained, since an individual is not trusted on his own. All of this applies when they saw the moon on the night of the thirtieth; but if they saw it before nightfall, at a time fit for sanctifying the month and for accepting testimony, no testimony is needed at all, since "hearing testimony is not superior to actually seeing."