There are two ways the times here can be worked out: by choosing a specific shul's zmanim, or by using your location.
When a shul is selected, the candle lighting and motzei times are taken directly from that shul's luach as published.
Earliest and latest times to light are calculated using the methods set out below. Lighting after the time your kehilla lights can be problematic — please speak to a Rov.
Responsibility for a shul's published zmanim remains with the shul.
Calculated zmanim should be treated as indicative rather than definitive.
The Gra: sunset less one and a quarter seasonal hours, where the day is measured sunrise to sunset and a seasonal hour is a twelfth of it.
It is always rounded up to the next minute, so that the time displayed is always after plag.
In location mode, candle lighting is sunset less 18 minutes.
Yerushalayim defaults to 40 minutes, in keeping with minhag Yerushalayim.
The usual time for lighting should follow minhag hamokom.
In location mode, motzei is calculated at 8.5 degrees below the horizon.
Some shuls calculate this differently — GGBH uses 7.083 degrees plus six minutes. As above, when a shul is selected the time displayed is taken directly from its luach.
Rabbeinu Tam is 72 fixed minutes after sunset.
Sunrise and sunset come from the NOAA solar position equations. These have been cross-referenced against ‘known true’ zmanim, and the greatest deviation found was 4.9 seconds.
It computes for a sea-level horizon. Elevation and other geographical features are not accounted for. Where the place you are in takes these into account in its own zmanim, that should be followed as minhag hamokom.
Where sunset or nightfall cannot be calculated — inside the Arctic circle, for example — no time is displayed. A Rov should be consulted.