The Greek name says what the bread is for. Anti means in place of and doron means the gift, and the gift in question is the sacrament. Antidoron is the remainder of the offering loaves after the Lamb and the commemorative particles have been cut from them at the preparation. It has been blessed and it has been part of the offering, and it is emphatically not the body of Christ, a distinction that Orthodox writers are careful to keep because the whole usefulness of the practice depends on it.
At the end of the liturgy the people come forward a second time, in the same order they would come for communion. The bread is received in both palms, the right hand crossed over the left, from the priest, and in most parishes the communicant kisses his hand as he gives it. It is eaten in the church before leaving rather than carried off casually, though taking pieces home for members of the household who could not come, and especially for the sick, is an old and encouraged custom. In many parishes the antidoron is eaten with a little wine and hot water at a table at the back, which turns the end of the service into the beginning of the parish meal.
The etiquette varies by local church and none of it is canonically fixed. What is fixed is the sequence: the sacrament first, to those prepared for it, and then bread to everybody in the building.