How the dates are estimated
The dates here follow the Umm al-Qura calendar of Saudi Arabia, which is computed in advance from the positions of the sun and the moon at Makkah. It is the most widely used published Islamic calendar, which is why most printed timetables start from it.
It is an estimate of the start, not the announcement. Each Islamic month begins with the new crescent, and the religious authorities of each country confirm the start of Ramadan themselves.
Moon sighting and a day's difference
In many countries a committee looks for the crescent on the evening of 29 Sha'ban. If it is seen, Ramadan starts the next day; if not, Sha'ban completes 30 days. Because the crescent is visible earlier in some places than others, and because authorities use different criteria, Ramadan can begin a day later in one country than in another. The same applies to Eid al-Fitr. On each city's timetable you can move the dates one day later to match your country's announcement.
How sehri and iftar are computed
Sehri ends at Fajr, when the sun reaches the method's dawn angle below the horizon (for example 18° or 18.5°). Iftar is at Maghrib, when the sun's upper edge sets. Many timetables also print Imsak, ten minutes before Fajr, as a precaution. The fast's length is the time from Fajr to Maghrib. Hourora computes these from the sun's position for each date and place, with the method usually published in each country, and shows them in the city's own time zone.
Why your mosque may differ by a few minutes
Local mosques and national authorities publish their own Ramadan timetables. They may use another method or a different precautionary margin, compute for another point in the city, or round differently. Differences of one to five minutes are normal. Your local mosque's timetable and the official moon-sighting announcement are the ones to follow.