Bazi comes in different names; the most common used names are Eight Characters (八字) , Four-Pillars (四柱), and Zi Ping (子平). Bazi is based on one's birth date and time to help determine the possible fate of the person. The birth date/time is first translated into chinese calendar representation which can be described by 4 pillars, where each pillar represents YEAR, MONTH, DAY and HOUR, with each pillar contains 2 Chinese characters (therefore a total of eight characters) , hence the naming of Eight Characters.
Bazi had long been used and practiced, but in Sung Dynasty, Xu Zi Ping, set the standard in using Bazi as the fortune telling tool.
There are 'two' main bazi fortune telling methods: (a) 神煞 and (b) 十神生剋制化. Xu's theory is based on (b) but many practitioners used (a) instead as it is much easier to use, but (b) is both much richer in its theoretical base as well as its application.
One should note that bazi only uses the birth time for fortune telling, but it does not include where the person is born and who this person's relationship, which also contributes a large portion to one's destiny. So bazi can only gives an indicator of a possible prediction. This is critical as many people think bazi or any fortune telling tool can be used to predict with 100% accuracy, I will leave this to you to have a thought about it.
The other analogy of Bazi is DNA: with the advance of genealogy, it is now possible to predict one's health condition and possibilites of getting certain diseases. Genealogy doesn't say one WILL certainly get some diseases but only suggests that one may be more susceptible. Bazi is similar whereby it can be used as a guide to what may happen to one's fate.