I was watching Harry Potter and the Half-Blood Prince but I didn't get why did Snape Kill Dumbledore?
This answer is one giant spoiler, and most spoilers are there in the open. Sorry, there's no way to do it without them.
Dumbledore's Death was pre-arranged by Dumbledore and Snape back in The Half-Blood Prince (starts at 0:54):
Snape had to kill Dumbledore. Firstly, because it was an act of mercy for Dumbledore, as he was already dying, first from the curse he received when he put on the Gaunt ring, and second when he drank the potion from the cave with the locket horcrux. That's what Dumbledore means when he says "Severus. Please.".
Snape also had to kill Dumbledore to remain a trusted Death Eater, and Voldemort's right hand. He needed to be a double agent in Voldemort's ranks in order to be able to help Harry & Co on their quest, such as when he sent his doe Patronus to the lake where he hid the sword of Godric Gryffindor.
As to how Snape became a double agent - it was after Voldemort had become determined to kill Harry as an infant, thus endangering his mother, Lily, whom Snape love since they were children.
Snape begged Dumbledore to protect Lily (see the video below); however, Lily still died. Snape continued to serve Dumbledore as a double agent to honour her memory (shameless plug), which was living in her son (and thus "You have your mother's eyes.")