After re-watching the movie, I realised that although Snape is revealed to be the 'Half-Blood Prince' - it is never explained as to how he got his name.
I assume 'Half-Blood' would be Half Muggle?
Anyone know the answer?
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As I recall, his father is a muggle, and his mother a witch. This would qualify him as half-blood under the blood purity classification. Additionally, his mother's last name is Prince. He would go by Half-Blood Prince because he values his mother's lineage over his fathers, possibly because if he claimed his blood status openly, it would be harder to use it against him, and perhaps because the whole sounds more mysterious or interesting.
This is explained in the book. Snape's mother's surname was Prince. Hence, he was a 'half-blood' Prince (not of royalty, just of the family Prince):
'I was going through the rest of the old Prophets and there was a tiny announcement about Eileen Prince marrying a man called Tobias Snape, and then later an announcement saying she'd given birth to a-'
'-murderer,' spat Harry.
'Well...yes,' said Hermione, 'So...I was sort of right. Snape must have been proud of being "half a Prince", you see? Tobias Snape was a Muggle from what it said in the Prophet.'
'Yeah, that fits,' said Harry. 'He'd play up the pure-blood side so he could get in with Lucius Malfoy and the rest of them.'
Half-Blood Prince Chapter 30: "The White Tomb"
(Quote courtesy of DavidS' Answer)