Finally is the same as lastly. It introduces the last element in a series.
- We must increase productivity. We must reduce unemployment. And finally, we must compete in world markets.
Finally can also suggest that one has been waiting a long time for something.
- She has finally got a job.
At last suggests the idea of impatience or inconvenience resulting from a long wait or delay.
- She has passed her exams at last.
- When at last they found him he was dead.
In the end
In the end suggests that something happens after a lot of changes or uncertainty.
- We made ten different plans for our holiday, but in the end we went to Goa again.
At the end
At the end refers to position at the end of something.
- A declarative sentence usually has a capital letter at the beginning and a full stop at the end.