How the times are computed
Each time is a position of the sun. Dhuhr is when the sun crosses the meridian (solar noon); sunrise and Maghrib are when its upper edge meets the horizon, allowing for refraction. Fajr and Isha are when the sun is a set angle below the horizon, at the start of dawn and the end of dusk. Asr is when an object's shadow equals its length (twice its length in the Hanafi school) plus its shadow at noon.
Hourora computes the sun's declination and the equation of time for each date and place (the NOAA and Meeus formulas), then solves for each angle, and shows the result in the place's own time zone, daylight saving included.
Why methods differ
Authorities chose different angles for dawn and dusk from their own observations: from 12° (France) to 20° (Indonesia, Singapore). Umm al-Qura, Qatar and the Gulf method put Isha a fixed 90 minutes after Maghrib. Several authorities also add precautionary minutes to their published tables; where the authority publishes them, Hourora includes them.
Why your mosque may differ by a few minutes
Local mosques publish their own timetables. They may use another method, round differently, add their own precautionary minutes, compute for a reference point elsewhere in the city, or set the iqamah later than the start time. Differences of one to five minutes are normal. Where your mosque's timetable differs, follow your mosque.
High latitudes
North of about 48° in early summer, the sun never sinks far enough for Fajr and Isha by angle. The usual rules then cap twilight at a portion of the night: the angle-based rule (the angle divided by 60), the middle of the night, or one-seventh of the night. Inside the polar circles, on days without sunrise or sunset, Hourora uses the times for latitude 65° and says so.