Doomsday is the name of a powerful supervillain and a deadly monster from the DC comics who is famous for being one of only two villains so far in DC history that have successfully killed Superman (despite the latter being resurrected afterwards).
The creature known as Doomsday probably does not mind Superman returning either since he too is an immortal with a truly terrifying adaptation - when he returns from death he becomes immune to whatever killed him before, this coupled with his already immense power serves to make him one of the most powerful and feared beings in the DC universe.
His name is derived from a comment made by the hero Booster Gold upon seeing the monster's rampage - stating it resembled "the coming of doomsday" - the media picked up on this comment and soon used it as the official name for the menace, Doomsday lived up to his name by destroying the Justice League without much effort and heading straight towards Metropolis - this alerted the attention of Superman who flew in to save the day only to find himself about to engage in one of the most memorable and brutal battles of his career.
Doomsday showed himself more than a match for Superman from the start and Superman became more aware that Doomsday would not stop until everything and everyone was dead - so he was left with no choice but to quite literally fight to the death, the two titans beating each other to death (although both of them later revived, as was inevitable).
Doomsday was a monster well before this event however and was responsible for the deaths of thousands across space/time before arriving on Earth in the original canon - though it was explained that Doomsday was, at least to begin with, a victim of another villain as he was created by a mad scientist to test immortality. Doomsday kept getting killed over and over again as part of this experiment and over the years grew to hate all life due to the continual torment he was caused throughout his long life, even killing his tormentor did not stop his anger and he kept on fighting.
This origin story also tried to make sense of why Doomsday fought so viciously against Superman by stating that he had instinctively learned to view Kryptonians as enemies due to these past experiments - Doomsday is rarely depicted as able to talk so this can't be confirmed fully, though it seems likely. Doomsday also has special mention due to the chaos caused in the aftermath of his original battle with Superman - which paved the way for Metropolis to be taken over by several fake Supermen (or a single clone in the animated movie, who also went insane).
Legends of the Multi-Universe: Armageddon
Doomsday appears as one of the final but main antagonists of this adventure.