In The Giver, what does "sixes" mean?

The sixes are children about six years old.

In Jonas’s community, all children born in a year are considered the same age.  There are only fifty children born each year, and they are assigned to parents when they turn One regardless of what month they were born in.  The Ceremony of One occurs in December, where the children get their names and families.

The Ceremony for the Ones was always noisy and fun. Each December, all the newchildren born in the previous year turned One. One at a time--there were always fifty in each year's group, if none had been released --they had been brought to the stage by the Nurturers who had cared for them since birth. (Ch. 2)

There is a ceremony for each age group up to twelve.  At the Ceremony of Twelve the children get their job assignments for life, and are no longer considered children so they no longer have an age number. 

There is really nothing special about the age of six.  The children are still young.

Fours, Fives, and Sixes all wore jackets that fastened down the back so that they would have to help each other dress and would learn interdependence. (Ch. 6)

Each ceremony includes a milestone and a gift, usually clothes like the Sixes' jackets.  At Seven, children get special jackets that button in the front, but up until then their jackets button in the back, which promotes interdependence because they need to help each other put them on.

The fact that the children of an age group are all assigned a number which designates birth order, but otherwise they are all considered the same age regardless of when they were born, contributes to the conformity and control that dominate the community.  They call it Sameness, which basically means that everything they do and every move they make is orchestrated by the community.