Buddhist nun
Specifically a female Buddhist monastic; the term is neutral and commonly used in news and general writing.
She became a nun at a young age.
About ten Buddhist nuns live at that temple.
Catholic nun; sister
Used for Christian nuns, especially in Catholic contexts. シスター is also common in everyday speech.
The nuns offer prayers in the chapel every morning.
She spent her whole life as a nun.
シスター is the common, casual loanword for a Christian nun, while 尼僧 is a more formal, written term used in the same sense.
修道女 is the formal Japanese term for a Christian nun, roughly equivalent to 尼僧 in its Catholic sense, while シスター is more casual.
尼僧 is a more formal, written term for a nun (Buddhist or Christian), while 尼 is the shorter, everyday word.
僧 refers to a male Buddhist monk, the direct gender counterpart to the female 尼僧.
修道士 refers to a male Christian monk, the direct gender counterpart to the female 尼僧 in its Catholic sense.
尼僧 shares the 僧 character with 僧侶 but specifically means a Buddhist nun, not a monk in general.
counter for people
nunnery; convent
尼 (nun) + 僧 (monk/priest). The kanji 尼 originally referred to a Buddhist nun, and 僧 is a general term for a member of the clergy. Together they specify a female monastic.