The Doublet System is the hostage exchange between the gods and the demons. Basically, with the market share battle between the gods and the demons, battles can get very heated and it can be very tempting to kill off your opponent. However, since it seems that gods and demons don't procreate very often, neither can afford any loss in headcount.
Thus was born the Doublet System. Basically, when a demon and a god(dess) are young, they meet in one room and link their souls. Their memories are then erased. The purpose behind this is that randomness of the system makes it risky to kill an enemy, since you might kill someone of your own.
According to Hild , she accepted doing this system for the sake of her daughter, Urd, who chose to be a Goddess.
Welsper and Belldandy are a pair in the doublet system, so if one dies, the other dies as well.