After her death, Yuri reached the 'afterlife,' and, having doubts about the 'afterlife,' she begins thorough investigation, meeting her first non-NPC friend, Hinata. She discovers that:
NPCs= Act like normal students, and they help the newcomers of the 'afterlife' feel and act like they're living a normal life. However, they apparently do not possess souls, and are merely 'programs.' Think of them as A.I. of games.
Non-NPCs= Those who died and were brought to afterlife. According to the current story, they were brought to the 'afterlife' because they had unsatisfied longings or wishes that they could not accomplish before they died.
The NPCs, by making non-NPCS live through ordinary, everyday life like before they died, would help them eventually accomplish the non-NPC's wishes or make them become satisfied enough to the life in the 'afterlife,' allowing them to pass on.
However, Yuri believes this to be disappearance of existence, therefore forms SSS to resist from being disappeared.
The angel, Kanade, clumsy as she is, figured out the real purpose of 'disappearing,' and decided to help the non-NPC students to pass on. She had somehow found out a way to manipulate the world by using a computer, thus finding a way to design weapons and super-human powers for self protection.
Later after Otonashi arrives, pretty-straight-forward-no-need-to-explain-things happen, and Iwasawa, having granted her wish to sing like she wanted to, disappears.
The Guild- the underground network of tunnels that lead to the factories where the members of the SSS receive their weapons. It is also operated the members of the SSS who do not want to disappear (although why they would want to be stuck underground, I wouldn't know)
Ayato Naoi- self-proclaimed God, he is a dumbass. Somehow, he had developed hypnotizing skill. He was a son of immensely famous potter; his older brother, who was a prodigy of potter, died, and Naoi was forced to take over his place. He was never accepted as a potter by his father. While he was supposed to disappear after Otonashi acknowledges him, he does not disappear and aid Otonashi in helping everybody pass on.
After Naoi arc, Otonashi becomes friend with Kanade. However, her cloning ability malfunctions, producing many other evil doppelgangers. She is then taken to underground. After the Seven Epic Sacrifices, Otonashi saves her; however, Kanade, having absorbed all the doppelgangers' different personality and experiences, faints, because her brain could not take it at that point.
When Kanade recovers, Otonash figures about the purpose of the 'disappearances,' and decides to help her.
In Episode 10, Yui passes on, having fulfilled her wishes of moving, wrestling, soccer, and marriage (although not formally completed).
Shadow-like beings= Created by an unknown person at this point, think of them as 'softwares that were infected with virus,' or simply put, zombies. By using computer, the perpetrator caused the NPCs to turn into the shadow-like beings. Their purpose seem to be making the non-NPCs who are not willing to pass on to become NPC, by (in my guessing) passing their souls on, or just consuming and destroying their souls. Just think of NPCs and Shadows as programs.