How does one pray Fajr and تحية المسجد (the two rak'ahs upon entering the mosque)?

If the questioner is referring to the Sunnah (definition) of Fajr, it consists of two rak'ahs performed after the dawn breaks. These are among the emphasized regular Sunnah prayers, and they can be performed in the mosque or at home, and they suffice for the "Tahiyyat al-Masjid" (greeting of the mosque) prayer. If the questioner is referring to the obligatory Fajr prayer, it consists of two rak'ahs from the five obligatory daily prayers. As for "Tahiyyat al-Masjid," it is a two-rak'ah prayer prescribed upon entering the mosque in a state of purity at any time, even during the prohibited times for prayer.