Basically the summary of the plot is that the japanese government has launched a brand new law: because of the lazyness and apathy of Japan's people it was decided that someone, daily and randomly, will be chosen and killed with a poison's capsule in order to "bring to life again" and wake up the others.
When kids get their vaccines at a young age in elementary school one in one thousand will be given a nano machine capsule that will have a set time to rupture. The date is predetermined (usually age 20-24) but random people are chosen.
The person who got one of those nano machine will be warned with an ikigami (or death's notice) when he/she will have only 24 hours left to live.
The main character of this story is Touhon. He was assigned the job of issuing out ikigami. Basically every month he is given the papers and he delivers them to the assigned people.
Each story in the manga deals with another person's back story and how they deal with the ikigami.
I am guessing that this system has been used for quite some time because society in Japan accept it naturally. If people try to go against it they will be institutionalized and tried to be trained to accept it or they will be given the capsule.
The main character will start to question this system as he meets with the selected people who get an ikigami.