First Wizarding War (1978-1981) Severus Snape eventually joined the ranks of Lord Voldemorts Death Eaters. His actions as a Death Eater are largely unknown, though he quickly became an important Death Eater in Voldemorts inner circle.
One may also ask, was Snape always a Death Eater? Yes, Snape was a death eater but more importantly, he was also a double agent. In our case, Snape was working for both Voldemort and Dumbledore and both trusted him - Voldemort thought that Snape was his most trusted death eater, whereas Dumbledore thought that Snape was loyal to only him.
Keeping this in view, why was Snape a Death Eater?
The only thing restricting a wizard from becoming a Death Eater would be if he or she was a Blood Traitor or a Muggle-born. Severus Snape proved loyal to Lord Voldemort when he overheard the prophecy and told it to the Dark Lord. Long story short, Voldemort trusted all his Death Eaters, no matter their blood status.
Why did Snape kill Dumbledore?
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. Thats what Dumbledore means when he says Severus.