General
Occasionally, when some people die, they become part of one of two sides: Reapers and Noah.
Reapers want to preserve the balance, while Noah want to destroy it.
They can choose to move visibly or invisibly in the world.
Because of their different goals, they've come into conflict at different points in history.
Timeline
More or less equivalent to early canon. Yeegar is dead and the Noah are on the move. Other Reapers are being sent out to find and aid the remaining Grand Reapers (the Generals).
Currently, there is no known equivalent to the Heart in this world; by attacking the Grand Reapers, the Noah are making a show of power more than anything, and the Order can't afford to lose their strongest Reapers.
Lost Souls
Most souls can move on immediately, but plenty need to collected by Reapers. No one knows why some souls need the help while others don't.
If souls are not collected shortly before or after death, they become lost.
Lost souls can still be collected (or destroyed), but it's more difficult, because they start causing supernatural incidents and usually resist collection.
The Black Order
A society of Grim Reapers and supporters. Also the location of the Path, and the repository for souls waiting to move on.
Headquarters actually refers to two buildings: the Limbo branch, where the main opening to the Path sits and only the dead can be, and the Main branch, where the dead and the living staff alike can work together (thanks to a special property of the area).
Other branches are usually only staffed by the living.
Grim Reapers
Grim Reapers are responsible for collecting and safeguarding the souls of the dead until it's time for them to move on.
Reapers are people who have died, but they're not "proper" dead: they're trapped between life and death. They do eventually die and move on. It is possible to kill them, but they're harder to kill than the living.
----- The Countdown
More often than not, Reapers can tell when a person is going to die, but only if they have a few days or less to live.
The countdown is imprecise -- more of an estimate. The less time is left, the more accurate the countdown is, and vice versa.
Countdowns don't always occur when they should. They sometimes occur when they shouldn't, in that the person will die but the soul will move on immediately.
Reapers cannot predict cause of death.
----- Scythes
"Scythe" is just the general name for a Reaper's tool that serves both as a weapon and something that can catch souls. (More or less equivalent to Innocence.)
Each Reaper's scythe is tied to their soul and can't be used by anyone else.
Scythes usually have multiple forms: a "normal" form, at least one weapon form, and an actual scythe form.
Scythe forms are used to collect souls.
"Weapon form" is a bit of a misnomer as scythe forms can also be used as, well, scythes. The powers unique to each Scythe can only be used in weapon form, though.
The Path
The Path is what allows the souls of the dead to move on -- for collected souls, a literal "heading toward the light". No one knows if that light is a way to heaven, or hell, or reincarnation, or something else. (The Reapers and other "in-between" dead haven't joined the "proper" dead yet, which is why they don't know.)
The Path isn't always open, which is why collected souls need to be safeguarded until it does.
Support Staff
The Order's support staff support the Reapers, maintain the Order, guard the collected souls, and study the Path. They also try to invent and improve on things that could help the Reapers perform their jobs.
There are two sections to support: the Living and the Dead. There can be some difficulties coordinating, of course, but overall they work well together.
The dead support staff are in a similar situation to Reapers in that they're neither living nor truly dead, but they don't have any special abilities they didn't have while living. They also can't be seen by the living without help.
The Noah
A group opposed to the Black Order. They're led by the Earl.
Similar to the Reapers, in that they're neither exactly living nor dead and can be killed. They also have powers, but these aren't tied to weapons.
They have Memories. When they awaken, they die and become Noah.
They're known by a few names. Some people call them demons, and others call them Snatchers. The name "Noah" refers to the time they upset the balance of the world so greatly that it caused the Great Flood.
The Earl
The leader of the Noah and creator of the Akuma. His plan seems to be to destroy the world by unbalancing it, but he hasn't exactly sent out pamphlets explaining things.
Akuma
Souls enslaved in mechanical bodies and forced to serve the Earl. They make the Reapers' jobs more difficult both by attacking them directly and by going on killing sprees, which creates an excess of lost souls that need shepherding. (This makes it difficult for Reapers to get all the souls.)
An Akuma can become strong enough to destroy souls on its own.