Budgies breed just like most other birds with the male rubbing his cloaca against the female. His sperm is then deposited onto the eggs, and the female then lays the eggs 10 to 14 days later.
Budgies need to be at least 10 to 11 months old before they are really mature enough to breed. It is a good idea to prepare them early by feeding them calcium supplements as the eggs require it in order to harden, and if the female does not have enough, she will start leaching it from her bones. Finally, they need a quiet and comfortable location.