the day in question; that day; the day of
Refers to a specific, previously mentioned or contextually understood day, such as an event, appointment, or deadline. Often used adverbially without a particle.
On the day of the exam, I arrived at the venue early.
チケットは当日会場でお求めください。
Please purchase tickets at the venue on the day of the event.
Please gather in front of the station on the day.
Please bring identification on the day.
Please reply within the same day.
There will be a fee if you cancel on the day.
同日 means 'the same day' as another day already mentioned, while 当日 means 'the day in question' (often the day of an event), and the two are easy to mix up because of their similar kanji and readings.
当時 means 'at that (past) time', while 当日 means 'on the day in question' (e.g. the day of an event), and learners often mix up the two 当- words.