Similar in concept to a Gigai, Regai are false bodies that allow for modified souls to have a physical form in Soul Society.
Modified souls have no usable physical form regardless of whether they are in Soul Society or the World of the Living. In the World of the Living they can be inserted into a Gigai, giving them a human form. Regai are used to give modified souls a human form while they are in Soul Society. Upon defeat, a Reigai returns to its mannequin-like form for a short period of time (usually seconds) before completely disappearing leaving only the spherical modified soul ball behind.
Unlike Gigai, Reigai are frequently altered and can be given abilities that extend beyond the capabilities of the modified soul within them. An example of this would be the Regai of Izuru Kira. The Reigai's Shikai form increases the weight of any object exponentially by a factor of ten per strike rather than two, the amount that the real Izuru's Shikai is able to do.
While visually identical under normal circumstances, Regai eyes have a bright blue tint that becomes more and more pronounced in a manner that is directly related to their current state of aggressiveness. During combat, their eyes actually produce visible currents of electricity that emanate from them.
Another difference between the two false bodies are that Reigai are far more aggressive and wiling to damage themselves in order to damage their opponents, unlike a Gigai which retains whatever level of aggression the modified soul naturally has. While some characters have attacks and abilities that they refuse to use for a variety of reasons such as pride or not wanting their powers known, the Reigai versions don't hold themselves to these restrictions and will use all revealed abilities of a character during combat.
Kageroza Inaba creates an army of Regai that house modified souls which are based off of various captains and lieutenants in the Gotei 13. The Regai are designed to be identical in appearance to the shinigami that they are based off of and Kageroza uses them to infiltrate the Gotei 13 and do his bidding.