My question comes from what I learned in the movies, considering I have not yet been able to read the books. I would just like to know why Severus Snape called himself the Half-Blood Prince? Did many people call him that or did he just think that it was a cool nickname? This question might be answered in the books so feel free to quote from it.
His mother's name was Eileen Prince, and his father was a Muggle (Tobias Snape). Thus, he gave himself the name "The Half-Blood 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"
It's never specified whether anyone else knew him by this name, but considering his Slytherin origin and the fact the Voldemort was on the rise, it's unlikely he was telling other student's about his half-blood status.
Well, maybe one other student...
Snape was a unpopular kid when he attended Hogwarts, and was always bullied by Harry's father. His mother also attended Hogwarts, and was a pure-blood with the last name "Prince". Snape's father was a muggle, so when Snape was born, he was a half blood. The kids who bullied him called him "The Half-Blood Prince", and that is what he was known as.
A mud blood (a really mean name) is one who is born from a non wizard family. A half blood is one who is born to one wizard and one muggle parent. A muggle is one who is not a wizard. Snape was a half blood, therefore the half blood part, and the prince part was from his witch mother, Eileen Prince, therefore the prince part, and his name, Snape, is from his not wizard father, Tobias Snape.