Late is when you do something after the time which you wanted to do it.
Later is when you do something after another thing.
Lately is when you do something not a long time ago. It is a synonym of recently.
- I was late for the dentist. My appointment was for 3 but arrived at 3:15.
- I will do it later, after I have eaten my dinner.
- Last week I went to a restaurant. It was the best meal I have had lately.
