The fajr prayer manifests itself with the first light in the morning and is usually defined by 18 degrees (astronomical twilight) below the horizon.
The true and false fajr
The islamic texts divide fajr into the true and a false fajr. The false fajr is called zodiacal light and comes usually below 18 degrees or in the very beginning of the astronomical twilight. Also the false fajr can only be seen all year in countries near the equator. In northern or southern countries it can be seen only for a small period in spring and fall and is absent in the other months. So the true fajr is defined by the first possible light at 18 degrees of astronomical twilight.
Fajr is at 18 degrees astronomical twilight
Different Islamic organisations defined and verified fajr at 18 degrees, which is scientifically the first possible light that can be observed. Among the organisations supporting 18 degrees is the Fiqh Council of the Muslim World League in Makkah. In Saudi Arabia fajr is at 18.5 degrees, Egypt defined Fajr at 19.5 degrees and some Asian countries defined fajr at 20 degrees.
On the other hand we see Islamic organisations in Europe and America who defined fajr at 12-15 degrees. Some of this organisations say that they use 12 degrees (France) just to make the fajr times easier while others claim that the true fajr is at this range of degrees. Historically Islamic scholars in the past defined fajr by 16-19 degrees. Those extreme methods do not have any foundations in long term observations and are not verified by Islamic scholars. While there is no harm to pray fajr after the beginning of it's time, a Muslim should avoid such methods which could potentially nullify his fasting especially in the month of Ramadan.
Fajr times in extreme latitudes (northern countries)
In countries above a latitude of 48 degrees in some nights in the summer, when the days get longer, the sun will not set below 18 degrees and the sun will not be deep enough for the sky to become absolutely dark. In such days the prayer times will be calculated by a special formula recommended by the Muslim World League.