nth time
Attaches to a number to express ordinal occurrence: 一回目 (the first time), 二回目 (the second time), etc. Used for counting repeated events or attempts.
This is my third attempt.
On our second date, we went to a movie.
This is my third attempt.
I finally succeeded on my third try.
Today is our second date.
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).
回 simply counts the number of occurrences (三回 = three times), while 回目 specifies which occurrence in order it is (三回目 = the third time).
Compound of 回 (かい, 'time, occurrence') and 目 (め, ordinal suffix).