A Blood Moon happens over the course of an entire night, which is 7:30PM-4:30AM in game time. A Blood Moon can occur if at least one player in the world has over 120 life (6 hearts) and it has a 1 in 9 chance (11%) of occurring on ANY night once the criteria is met. Blood Moons are signaled by a message "The Blood Moon is rising..." They also seem to occur more frequently in more evolved worlds (Hard Mode worlds, more players, higher health, etc.)
During a Blood Moon, enemy spawns increase, even near your Home and NPCs (where enemies do not normally spawn), Zombies and other enemies are also able to open Doors. Blood Moons also cause NPCs to sell different items.
You only need to complete the Blood Moon event for the achievement, which ends at 4:30AM in the game (the start of a day and the end of night). You do not need to stay alive the entire time and you can do it in any world. This means you can leave the game running overnight and get the achievement in theory (assuming a Blood Moon occurs at some point).
FASTEST SOLUTION
If a Blood Moon occurs early on in the game, you can simply bury yourself/close yourself in a small box of stone or other material and wait it out. Things can only reach you if they can pass through walls, like Wraiths.
You can also place blocks behind a door or replace your doors in any house with wall material to prevent enemies from getting in by opening doors if you want to keep yourself or your NPCs safe.