What is the justice system of the Ibo culture in Things Fall Apart before the white men's arrival?

The Ibo justice system before the white man's arrival was very traditional and intertwined with their religion. For example, the many taboos and forbidden actions could be counted among laws. The novel points this out several times that there were some aspects of the lives of Ibo people that didn't need to be enforced at all because breaking those rules was unthinkable. This includes one of the crimes committed by a Christian convert, who kills the sacred python, the emanation of the god of water. The novel has this to say on the matter:

If a clansman killed a royal python accidentally, he made sacrifices of atonement and performed an expensive burial ceremony such as was done for a great man. No punishment was prescribed for a man who killed the python knowingly. Nobody thought that such a thing could ever happen.

As another answers to this question have pointed out, the Ibo justice system revolved largely around balancing. If the natural order of things was pushed out of balance, it had to be restored. Everything and everyone in Ibo culture had their place and most of it was strictly immovable, emphasizing the idea of tradition and continuity.

Several times, the wiser men are shown to wonder whether all of their laws and customs are just, but before the arrival of the Christians, no one tries to change anything. The Ibo believed that the gods had made the world a certain way, and their duty was to maintain it. That included little crimes and big crimes, which could be set right, but also things that could not be fixed—like being born an osu, an outcast, who could not mingle with anyone but their own kind.

What is the justice system of the Ibo culture in Things Fall Apart before the white men's arrival?

The core ideal of traditional Igbo justice is founded not so much on retribution but on restitution. Igbo justice attempts to create balance and to right wrongs instead of just punishing the perpetrator. The idea is to ensure that justice also ensures recompense for the victim as much as meting out punishment to the criminal.

There is a distinction between divine laws and laws made by humans. The breaking of divine law results in severe punishment (these would for example be crimes such as murder and incest - these are taboo). Transgressing man-made laws (such as theft) would result in less severe punishment.

Punishment for serious offences could result in execution, banishment or permanent compulsory exile. In this instance, restitution to the victim's family could be in the form of awarding the perpetrator's land and property to them or 'a life for a life'. 

Punishment for lesser crimes involved the criminal either being ostracized or having to pay compensation to the victim. The criminal might also be publicly humiliated and held up for derision by the villagers.

Traditional courts comprised the elders of the village (the egwugwu, comprising men only) who would then consult with the gods to establish punishment once a guilty verdict had been reached. The 'audi alteram partem' rule is applied. Both parties were given the opportunity to present their version of a matter. There was no formal court building or complex, and trials were conducted openly. Both parties had to undertake an oath in which they promised before their gods that they would tell the truth.

Based on the evidence presented, the egwugwu would then, after consultation with the gods, decide on the degree of illegality (the severity of the crime) and would decide on compensation or mete out punishment.

Okonkwo for example, was instructed to pay a fine after dishonouring The Week of Peace by beating his wife, Ojiugo, for not preparing him a meal. He was later banished to his mother's village, Mbanta, for seven years for accidentally shooting and killing one of Ezeudu's son during Ezeudu's funeral. 

What is the justice system of the Ibo culture in Things Fall Apart before the white men's arrival?

The Ibo have a complex justice system, that involves all members of the community. The role of judges are played by the egwugwu, who are prominent citizens of the village wearing masks. The masks represent the ancestral spirits of the village, who pass judgment upon the accused.

Each person who brings a suit to the egwugwu gains a trial, during which both sides plead their cases, much like prosecutors and defenders in contemporary legal courts. After hearing the case, the judges will confer together, then decide the best course of action. Often, if the case warrants punishment, it will be a very public one, usually carried out by all members of the village.

All this changes when the missionaries arrive. Suddenly, the Ibo are subject to laws which are not their own, & punishments that serve no purpose other than domination and humiliation. In fact, the treatment of the Ibo in the white legal system plays a large role in the final events of the novel.  

What is the role of justice in Igbo society in Things Fall Apart?

One of the things that Chinua Achebe did in the novel Things Fall Apart was portray the nuanced way that crime and punishment worked in Igbo society. Many of the customs seemed to be primitive or arbitrary to other observers but Achebe showed the way that the "gods" worked to resolve conflicts and administer justice to villages and between villages.

In this way, justice was used as a way not only to resolve present conflicts but also to build connections between villages and to redress old wrongs. When Okonkwo killed one of his wives with his terrible gun, it could have been disastrous for him to be killed or jailed, so it made sense to banish him so that the village sees that he is punished but he eventually has the chance to return and rebuild what he lost. 

Achebe also demonstrates perhaps that justice was sometimes flawed and particularly when Okonkwo goes against his friends' advice and takes part in killing Ikemefuna, the resulting trouble demonstrates some of the flaws in the justice system of the village.