What’s the most sacred part of the mosque? How does the mosque get its name?

Allow me to answer your questions in reverse order. 

The term mosque comes from an old Arabic word meaning to worship, especially in prostration. This term, sajada, is modified by the prefix ma- (meaning place) into masjid- a place for worship. This term was changed through contact with European cultures like the Spanish (mezquita) and Italians (moscheta.) The immediate predecessor to the English pronunciation of mosque was the French mosque. In may parts of the world, a mosque is alternately called a masjid. These two terms really mean the same thing, but mosque has a long history of cultural interpretation behind it. 

Inside of a mosque or masjid, the entire place is considered to be similarly holy or owed reverence as any other house of worship. The interior may be designated into spaces for ritual ablution before prayer, study, conversation/social gathering, and worship. Some parts of the mosque may be specially designated according to Sunnah (religious tradition, almost equivalent with law) to be used only for prayer. Technically, this part of the mosque is specifically the masjid because it is the place only for worship. This may be considered the most sacred part of the mosque, though an equally appropriate answer would be the mihrab. This is a special nook in the wall of a mosque which designates the direction of Mecca. The mihrab is special because it offers a more concrete and visual connection with the holy city of Islam and the location of the kabba.