The criterion of deciding the beginning of a Hijri month
Question

Some people claimed to have sighted the new moon of Ramadan, whilst astronomers and those who follow calculations claim that it is not possible to have sighted it on that night. There is no confusion in my mind because of this, as calculations may be wrong. But what does confuse me is that the astronomers and those who deal with calculations claim that they tried to sight the moon that night with the telescopes and other equipment that they have but they did not see it. How can it be seen with the naked eye and not with modern equipment? If it were the other way around, and they saw it with their equipment but it was not seen with the naked eye, there would be some justification for the difference of opinion as to whether we should fast or not, and whether the people should break the fast or not. But the problem is: How can they see it with the naked eye and it is not seen with their equipment? In fact I would like you to explain clearly so as to dispel my anxiety and confusion, because I do not think I am alone in this matter.
Praise be to Allah.How to confirm the beginning and end of Ramadan
What counts with regard to confirming the beginning of the month of Ramadan is sighting of the new moon or the completion of 30 days of Sha`ban if the moon is not sighted. This is what is indicated by the authentic Sunnah, and the scholars unanimously agreed on it. 
Al-Bukhari (1909) and Muslim (1081) narrated that Abu Hurayrah (may Allah be pleased with him) said: The Prophet (blessings and peace of Allah be upon him) said: “Fast when you see it (the new moon ) and break the fast when you see it, and if it is cloudy then reckon the month as thirty.”  
Do astronomical calculations count in determining the beginning and end of Hijri months?
Astronomical calculations do not count . The basic principle with regard to sighting is that it should be done with the naked eye, but if the crescent moon is sighted with modern equipment , then this sighting may be acted upon. 
With regard to how the new moon is sighted with the naked eye when it is not seen with telescopes and equipment, this may be due to differences in the place and time of sighting. 
Whatever the case, the ruling depends on the sighting of the new moon. So long as it has been seen by one or two trustworthy Muslims, then it is obligatory to act upon this sighting. 
Shaykh Salih ibn Muhammad Al-Lahaydan (may Allah have mercy on him), the head of the Higher Judiciary Council (may Allah preserve him) said: 
“There is a brother called `Abdulah Al-Khudayri who is famous for sighting the moon and he is one of those who are known to be involved in sighting the moon at different stages, even at times other than the new moon. Some astronomers went to him and met with him in the region of Hawtah Sadir. He told me that they estimated that the moon would appear on that night in a certain place according to the calculations they did on their computers, but he told them that it would not appear from the place that they said, because he had seen it before them the night before, and he knew the stations of the moon and from where it would rise every night. Then when the moon appeared, it rose from the place he mentioned and not the place they mentioned. However, he pardons them because  their conclusion was not based on direct moon sighting; rather it was based on calculations with the computers they had.” (From an interview with him in Ar-Riyadh newspaper) 
For more details, please see this category: Seeing the New Moon.
And Allah knows best.