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Jewish days nominally start and end at sunset. Certain mitzvot can be observed only during the day, and others during the night; sunrise is nominally taken as the dividing point. The twilight times after sunset and before sunrise are of somewhat questionable status as to which date they are part of, and whether they are day or night. The most well-known consequence of this questionable status is that Havdalah marking the end of Shabbat takes place not at sunset, but when three stars are (or would be) visible. However, there are less well-known effects on the timing of many other mitvot, in particular the zmanim - the times by which certain prayers must be completed or before which others cannot be started.
Two particular times are considered important in this respect. Alot hashachar is the time of the first light of dawn. Tzeit hakochavim is nightfall, the time when the stars emerge. There are two main traditions for alot and many for tzeit.
Alot is generally taken as 72 minutes before sunrise; others treat it as 90 minutes. However, 72 minutes at one latitude at a certain time of year is much different in regard to sunlight than 72 minutes at a different latitude or a different time of year. Since it is the amount of sunlight that really determines alot, it becomes more meaningful to standardize using the sun's elevation 72 minutes before sunrise, with the standard location being Jerusalem and the time of year being the equinox. For that date and location, the sun is 16.1° below the horizon 72 minutes before sunrise, therefore 16.1° is used for all locations and all dates. This elevation is known as 72 equivalent minutes. Similarly, 19.8° is 90 equivalent minutes. Nevertheless, others use 72 or 90 civil minutes (the unchanging fixed minutes which we use in everyday life) to determine alot.
As mentioned above, tzeit has many different interpretations. One of the commonest is 24 equivalent minutes (5.95°); however, Rabbenu Tam felt that it should be 72 equivalent minutes, or, according to some, 90 equivalent minutes; in either case the number of degrees is the same as at dawn. There are minority opinions for 18.6 equivalent minutes (4.81°), 120 equivalent minutes (26°), 13.2 (3.5), 27.8 (6.5), and 66.6 (18). One problem with all of these measurements is that they are either, like 5.95°, slightly too soon for 3 stars, or, like 16.1°, much later than necessary. Those who use the late calculations will certainly mark havdalah at tzeit; however, the earlier calculations require an extra determination for havdalah. The commonest value here is 42 civil minutes after sunset (24 for tzeit and 18 as with candlelighting time before Shabbat). Many others use 72 civil minutes. And there are, of course, various elevations used as a substitute for the fixed minutes - 8.5 or 8.75 degrees for 3 small stars, and 7.5 degrees for 3 medium stars.
Even if there were universally accepted definitions for alot and tzeit, there are two main systems for calcluating zmanim. The Vilan Gaon, the GR"A, treats day as starting at sunrise and ending at sunset. Magen Avraham treats day as starting at alot and ending at tzeit. In either case, the period of daylight is divided into 12 portions known as shaah zmanit or proportional hours. Earliest tallit is commonly taken as an elevation of 11°; others use 10.2° or 50 or 60 civil minutes. The sh'ma must be said within 3 proportional hours from the start of day (sof zman sh'ma); the amidah within 4 (sof zman tefillah). Chatzot is the exact midpoint between start and end of day. Mincha Gedolah, the earliest permissible time to recite Mincha, is the greater of 30 civil minutes or 30 proportional minutes after chatzot; this is sometimes referred to as 30 strict proportional minutes (note that some always use 30 civil minutes or 30 proportional minutes for this calculation). Mincha Ketanah, the preferred time for Mincha, is 9.5 proportional hours after start of day. Plag Hamincha, 10.75 proportional hours into the day, is the earliest permissible time to start the new day (by bringing in Shabbat or reciting Maariv).
These calculations are, of course, approximate, and some padding should be used if you intend to make use of them. They will certainly not work for far northern or far southern latitudes at certain times of the year, when the sun does not rise and/or set during the day (or rises very early and sets very late); since I am not sure what is halakhically required in those cases, I will have to leave it to the "customer" to figure out what's right. Likewise, there are some surprising opinions about Shabbat in the Far East and West which are not reflected here.