Cockatiels are some of the easiest birds to get to breed if you do it right. It's easy for anyone to put two cockatiels together and watch as eggs are laid and hopefully chicks are born. However, there's a lot more you need to know about cockatiel breeding.
Puberty in cockatiels happens around the 18 month old mark. Although it isn't strictly puberty, it's normally what cockatiel breeders call the period from when they hatch until they become "safe to breed". If you mate cockatiels before they are 18 months old, especially if the female is under 18 months, then complications could occur such as defect chicks could be born or even one of the parents could get sick from stress.
You must also be aware of the cockatiel's relation to each other. As in humans, if inbreeding occurs then the offspring will be more susceptible to diseases, may be born with defects and a whole range of other "nastys". If you're in doubt about whether the two parent cockatiels are related, then to avoid any stress, just opt to get a different cockatiel if you really want to breed them.
Once a male and female cockatiel are together, then its just a matter of time before they will start to mate, which will leave the female with some eggs to lay. Once she has eggs, she will lay them every once every two days and are incubated for around 3 weeks. It is important that the eggs are not touched until the mother has stopped incubating them which is normally 4 weeks after laying.
Some females also lay eggs even is she has not been mated, and so the eggs she lays, like chickens, are infertile. To stop her laying more eggs simply keep her in the dark for longer periods. E.G put a dark towel over the cage for two hours after you get up.