Harry spent the night before his birthday laying on the cold floor of a shack that was built on top of a large rock, way out to sea, watching his watch tick over until it hit midnight and became officially his birthday. And once it did, he got a visitor that would change his life for ever.
How do we know that Harry was born on July 31? This has been established in many places in the books, perhaps most importantly in Dumbledore's discussion of the prophecy in OP37. However, we can first determine this from PS8.
In the Daily Prophet, Harry reads:
"GRINGOTTS BREAK-IN LATEST -- Investigations continue into the break-in at Gringotts on 31 July..."
He reacts in surprise:
"Hagrid!" said Harry, "that Gringotts break-in happened on my birthday! It might've been happening while we were there!"