Twins are not allowed because having two identical people walking around would be confusing. The babies are weighed as soon as they are born, and the lighter one is euthanized (“released”).
In The Giver, everything is tightly controlled. The Community is ruled by a concept named Sameness, which basically means that everyone is the same. Choices are made for people. The goal is to ensure that no one is uncomfortable. For a person to be uncomfortable is a terrible thing in the community.
"Well, they can't have two identical people around! Think how confusing it would be!" Jonas chuckled. (ch 19, p. 146).
The smaller twin is released, which means that it is euthanized by lethal injection.
I'll have to select the one to be nurtured, and the one to be released. It's usually not hard, though. Usually it's just a matter of birthweight. We release the smaller of the two." (ch 15, p. 114)
When Jonas finds out what release really means, he is horrified. He cannot believe his father would kill an innocent newborn. As a result, he decides to leave the community, returning the memories to the people and ending sameness.
Lowry, Lois (1993-04-26). The Giver (Newbery Medal Book). Houghton Mifflin Harcourt. Kindle Edition.