The reason why Ramadan changes every year in relation to the Gregorian calendar
Question

I hope that you can explain with regard to the Hijri calendar: why does Ramadan come thirteen or fourteen days later than the Gregorian date?
Praise be to Allah.Firstly: 
It is well-known that different nations and peoples use different calendars. There is the solar calendar, the beginning and end of which are based on the movements of the sun; it has 365 days. 
And there is the lunar calendar, which is based on the appearance and disappearance of the moon at the beginning and end of the month; it has 354 days. 
The solar calendar is similar to the lunar calendar with regard to the number of months, but it differs from it with regard to the number of days. So it is eleven days longer than the lunar calendar. 
The Gregorian calendar is based on the solar year, whereas the Hijri calendar is based on the lunar year. It is for this reason that the beginning of Ramadan every year differs in relation to the Gregorian calendar, and as a result of that it moves through the four seasons. 
Secondly: 
The lunar calendar is the calendar that must be followed, because Allah, may He be glorified and exalted, says (interpretation of the meaning): “It is He Who made the sun a shining thing and the moon as a light and measured out its (their) stages, that you might know the number of years and the reckoning” [Yoonus 10:5].
Ibn Katheer said: From the sun the days are known, and from the phases of the moon the months and years are known.
End quote from Tafseer Ibn Katheer (4/248). 
The sun defines day and night only; as for the moon, Allah – may He be glorified and exalted – has made it a measure for the months and years, as He says: (interpretation of the meaning): “... and measured out its (their) stages, that you might know the number of years and the reckoning” [Yoonus 10:5]. 
And Allah, may He be glorified and exalted, says (interpretation of the meaning): “Verily, the number of months with Allah is twelve months (in a year), so was it ordained by Allah on the Day when He created the heavens and the earth; of them four are Sacred, (i.e. the 1st, the 7th, the 11th and the 12th months of the Islamic calendar). That is the right religion, so wrong not yourselves therein” [at-Tawbah 9:36]. 
The words “That is the right religion” indicate that this calendar is the correct way  that Allah has approved for us, and that anything other than it, that is customarily used by other nations, is not correct, because it is subject to mistakes and confusion. 
Al-Qurtubi said: This verse indicates that what is required is to connect the rulings on acts of worship and other matters to the months and years as known to the Arabs, and not the months that the Persians, Romans and Copts follow.
End quote from al-Jaami‘ li Ahkaam al-Qur’an (8/133) 
Ash-Shawkaani said: This verse states that no attention should be paid to the months used by the Persians, Romans and Copts, some of which they make thirty days long, and others they make more or less than that.
End quote from Fath al-Qadeer (2/521). 
Allah, may He be exalted, says (interpretation of the meaning): “They ask you (O Muhammad SAW) about the new moons. Say: These are signs to mark fixed periods of time for mankind and for the pilgrimage” [al-Baqarah 2:189]; i.e., they are signs for the people with regard to entering and exiting ihram [for Hajj]; for beginning and ending the fast; for marriage, divorce and ‘iddah; for dealing, trade and debts; for both their religious and worldly affairs. 
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: Here Allah tells us that they are signs to mark fixed periods of time for mankind. This is general in meaning and applies to all their affairs. Thus Allah has made the new moon signs to mark fixed periods of time with regard to the established rulings of Islam… That includes the fast, Hajj, ‘iddah, and expiatory fasts.
End quote from Majmoo‘ al-Fataawa (25/133). 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: As for the Western calendar, there is no tangible, rational or religious basis for it. Hence you will find some of the months having twenty-eight days, some having thirty days, and some having thirty-one days, with no known reason for this difference. Moreover, these months have no tangible sign that people may refer to in order to work out times, unlike the lunar calendar which does have a physical sign that is known to everyone.
End quote from Tafseer al-Baqarah (2/371) 
And Allah knows best.