Sai-in (西院,, さいいん) is the western precinct of an area or structure (such as the side of a temple complex). It is the opposite of Tō-in which refers to the eastern precinct.
The -in Suffix is often attached to words referring to well organized structures, such as in the Japanese word for hospital, Byōin.
The word Sai is one of two possible pronunciations for the Japanese written kanji character meaning west.
References
Sai-in (Western Precinct) Wikipedia(Text) CC BY-SA