By a Half Hour?

: 30 minutes I waited for half an hour.

What does on the half hour mean?

: at half past the hour : at 12:30, 1:30, 2:30, etc. Tours start on the half hour.

Do you say an half hour or a half hour?

"half-hour" is commonly written with a dash (hyphen) and is a distinct word in the dictionary. So, "a half-hour" = 30 minutes = "half an hour". In Australia we say 'half an hour'. You might hear 'a half hour', but it would sound stilted.