This sounds like it is from tax avoidance law so if you are considering using it maybe get it explained by someone from a legal background.
Election means choice.
If you have a married couple, and one earns a lot of money and the other earns a little, the higher earner pays more tax. Charitable donations are tax free up to a certain amount. If the higher earner wants to make a large donation of money to a charity and avoid paying any tax, they can choose to claim that half of the gift was given by the lower earner. In this way the couple overall benefits by paying slightly less tax. This isn't a standard procedure which is why it is a special election. The third party is the charity receiving the money.