'Nightly', 'every night', and 'each night' all describe something that happens once per night. 'Every night' is the most common in everyday conversation ('I brush my teeth every night'). 'Nightly' sounds slightly more formal and is common in writing or scheduled announcements ('the show runs nightly'). 'Each night' emphasises every single night individually, often used when describing a routine that repeats without exception ('each night she checks the doors'). 'At night' refers to the general time period rather than frequency ('it gets cold at night'). Use 'nightly' or 'every night' for frequency; use 'at night' to describe when something normally occurs.
'Nightly' comes from Old English 'nihtlic' (adjective, 'of or relating to night') and 'nihtlice' (adverb, 'by night, every night'). The word combines 'niht' (night) with the suffix '-lic' (which later became '-ly'). The '-ly' suffix here serves two functions: it turns the noun 'night' into an adjective meaning 'happening every night', and also into an adverb meaning 'on each night'. The word has been used in English since before the 12th century and has stayed close to its original meaning.