Late is an adjective and an adverb, describing an event that happened after the correct time:
We had a late breakfast at 10:00.
(the usual time for breakfast is earlier, around 7-9 AM)
The bus arrived thirty minutes late.
(the bus arrived thirty minutes after the correct time on the schedule)
Avoid this common error: if you arrive late for an appointment, don't say "Sorry for late" or "Sorry for the late." The correct expression is "Sorry I'm late."
Delay is a noun and a verb, describing a cause for something to be late:
We were delayed by heavy traffic.
(the traffic caused us to be late)
The delay in the project was caused by a miscommunication.
(the miscommunication caused the project to fall behind schedule)
Postpone is a verb - it is when you choose to do something at a later time than planned. Usually "delay" involves factors beyond your control (like the traffic or weather), whereas "postpone" is a more conscious decision to change the schedule.
Three team members aren't available on Tuesday - let's postpone the meeting until Thursday.
My daughter got sick, so we postponed our camping trip until the following weekend.