Because I have no photoshop skills.

Eirei system explanation from the Fate route.

Normal eirei.

Is what was mentioned in the Fate route. However, from Heaven's Feel and Unlimited Blade Works we know SOMETHING comes back.

"Clone" or the word bunshin probably ought to be better thought of as emanation (working under the Buddhist concept/excuse).

Arthur's case.

If Arthur gets the grail, she goes back into her own time, dies, and is relocated to the throne as a counter guardian. If Arthur fails, she goes back to her own time and then is sent to a different era for another chance at the grail. If Arthur gives up on getting a grail, she still goes back to her own time to die (and will probably still become a different sort of eirei).

So basically in summary....it all comes down to the world pact.

The usual path for becoming a hero is to make a pact with what's called the "World" while he's still alive. In exchange for power, fame, and fortune, the hero gives his afterlife to the World. It's a fair deal for the most part. Both sides get something.

Sometimes there are people that are powerful enough that they didn't have to make a pact in order to become a hero. People like King Arthur for instance. In her case, she made a pact with the World while on the brink of death (after she had already become a hero) to be able to get the Grail while still alive and use its power to undo her past. So, she was cycled to different eras with the opportunity of obtaining a Grail (but only participated in the Fuyuki ones during the 4th and 5th times). When she forfeited her desire and accepted her past, the pact was nullified and her being cycled through time was finally ended. She returned back to her dying body in her own era.

Archer, while alive, didn't have that sort of power. And in desperation just to save a measely few hundred lives, he was able to make a pact. A person can't be called a hero by just saving a few hundred lives, but in this case, what mattered hadn't been the numbers, but the fact that he saved lives that were fated to die. Overcoming destiny.

In either case, offering your afterlife means the soul, instead of being dispersed completely and returning to Akasha to be recycled, is freed from normal transmigration and enters one of the higher tiers of phenomena. If all phenomena originate and return to Akasha, then these souls are located on a slightly lower plane.

How do you make a pact? Who knows. Probably similar to the whole deal about contacting the Apostles in Berserk.

And then there are other people who didn't make any pact at all and didn't do anything significant but were raised up to the rank of eirei because of the one-sided and self-centered wishes of others. The person that became Angra Mainyu is basically a guy that was reduced to 100% wishes and 0% person.

Making a pact probably will mean being used as a Counter-Guardian after death (based on how Archer said that Saber hadn't become one yet in the HF route).