Late is both an adverb and an adjective; it means the opposite of early. Lately is also an adverb; it means ‘recently’.
Late meaning ‘not on time’
Late as an adverb means ‘not on time’:
Well, I couldn’t find my classroom, so I got to the classroom a bit late and then I had to sing a song in front of the other students!
I got to the classroom a bit lately…
The programme started late.
The programme started lately.
Late as an adjective means ‘after the usual time’:
We had a late breakfast.
describes when the activity happened
describes the train
Lately meaning ‘recently’
We use lately for states or for repeated events, mostly with the present perfect:
She says she’s been feeling tired lately. I think she’s working too much.
He’s been studying hard lately. He’s got exams at the end of the month.
We don’t use lately for single events that happened recently:
I had a strange experience recently.
I had a strange experience lately.
We don’t use lately as an adjective:
We usually have a late lunch.
a lately lunch.