Half of a dozen is a group of six items of any type. A dozen is a group of twelve items of any type.
Eggs often come in a half-dozen container, a full dozen container, or even with two dozen per pack. A baker's dozen is different than other types of dozens, since this refers to 13 items instead of the usual 12. A baker's dozen often has 13 items instead of 12 because of the likelihood of broken or oddly-shaped foods that the baker would want to give away. In older books and movies, characters shopping in markets often receive baker's dozens.