nth; number ... in order; ...-th in a sequence
Ordinal counter used after a number to show position in a sequence: 一番目, 二番目, 三番目, etc. It is more explicit about order than plain 番.
See also: 目
He arrived third.
division of a kabuki performance
Kabuki-specific noun for divisions or sections of a performance; this is a technical use and is not the ordinary everyday meaning.
She is the second child.
Please turn right at the second corner.
In kabuki, 番目 is used as a term for a division of a performance.
What number are you in the line?
K2 is the second highest mountain in the world.
第 is a formal prefix attached before the number (第二章), while 番目 follows the number and is the everyday way to say 'the nth' (二番目の章).
目 attaches directly after a counter to mark order (三回目, 一つ目), whereas 番目 specifically follows a bare number to state its position in a sequence.
回目 marks the ordinal number of a repeated event or attempt (二回目 = the second time), while 番目 marks the ordinal position of items in a sequence (二番目 = the second one).
Formed from 番, meaning number, turn, or order, plus 目, a common ordinal suffix. This is a practical component explanation rather than a detailed historical derivation.