Who gave the Invisibility Cloak to Harry Potter?

The Cloak was one of the three Hallows, and originally belonged to Ignotus Peverell, the youngest and, as it turned out, the wisest of the three brothers:

And then Death asked the third and youngest brother what he would like. The youngest brother was the humblest and also the wisest of the brothers, and he did not trust Death. So he asked for something that would enable him to go forth from that place without being followed by Death. And Death, most unwillingly, handed over his own Cloak of Invisibility.
_{Harry Potter and the Deathly Hallows, chapter 21: "The Tale of the Three Brothers"}

It passed through generations to the descendants of Ignotus, and ended up with James Potter:

‘The Cloak, as you know now, traveled down through the ages, father to son, mother to daughter, right down to Ignotus’s last living descendant, who was born, as Ignotus was, in the village of Godric’s Hollow.’ Dumbledore smiled at Harry.

‘Me?’

‘You. You have guessed, I know, why the Cloak was in my possession on the night your parents died. James had showed it to me just a few days previously. It explained much of his undetected wrongdoing at school!’
_{Ibid, chapter 35: "King's Cross".}

Dumbledore had borrowed it from James to examine it, and discovered, much to his surprise, that it was one of the Hallows. James and Lily died shortly afterwards, and the Cloak remained with Dumbledore.

He then passed it on to Harry, as h was its rightful owner (we thank BCdotWEB for pointing to this answer on SFF; some of these quote also appear in the Cloak's article on Pottermore):

Harry pulled off the Cloak and seized the letter. Written in narrow, loopy writing he had never seen before were the following words:

Your father left this in my possession before he died.
It is time it was returned to you.
Use it well.
A Very Merry Christmas to you.
_{Harry Potter and the Philosopher's Stone, chapter 12: "The Mirror of Erised"}

While it was not clear from the letter that it was Dumbledore, he acknowledges that later in the book:

When he had found his voice again, Harry said, ‘And the Invisibility Cloak – do you know who sent it to me?’

‘Ah – your father happened to leave it in my possession and I thought you might like it.’ Dumbledore’s eyes twinkled. ‘Useful things … your father used it mainly for sneaking off to the kitchens to steal food when he was here.’
_{Ibid, chapter 17: "The Man With Two Faces"}