The reigai are just pills like mod souls little tiny pills. If you watched one episode in the regai arc when you kill or beat a reigai it turns into a pill like when prople use kon's pill excepet the pill is a red instead of green or something. The other thing is that when you get poisoned,hurt,or about to die like what happened to Kageroza Inaba. When Mayuri Kurotsuchi disguised himself as his own reigai and stabbed Kageroza and poisoned him he killed the Isane Kotetsu reigai and took the pill so he didn't have to go through that. The poison that he used made him go insane he didn't have any control at all of his actions.
The reigai are two times stronger than the original shinigami. But without them as Kageroza said they are much stronger without the braclet or seal.
Kageroza Inaba was the lead scientist for project spearhead (the experiment that kon was in.),an initiantive in which the Soul Society experimented with creating artificial souls known as modified souls (mod souls) and implanted them into shinigami corpsesfor the purpose of increasing the battle capabilities of the gotei 13. Kageroza's research and experimentation
eventually led him to not only copy the mental characteristics of captains and implanting them in the corpes,but to enhance their abilites as well.