In a male dog, puberty appears around 8 or 12 months of age. Upon reaching purity, the dog will start to mark his territory (outside the , that is) with urine, stopping at every tree and a corner to do so.
In the female dog, the first estrus cycle appears around 7 or 8 months of age, and is characterized by a vaginal blood flow which can last for about 10 to 12 days and is followed by ovulation (fertile period). This cycle repeats every six months -- the regularity starts to disappear when the female dog is around 9 or 10 years old. If you don't want your female dog to ever procreate, it is advisable to sterilize her before she has her first heat, as this way the risk or probability of future mammary tumors is almost null. In case you don't want to opt for surgery, you can administer some medications in order to avoid the appearance of the estrus cycle, but you have to be informed about the possible undesirable secondary effects. For this reason, the use of such medications must be temporal (not throughout the animal's entire life).
You can also try other alternatives like male sterilization (castration or vasectomy). Making the choice between the different options really depends on every particular case.