There are a number of conflicts in the story, one of the main ones is the desire that Okonkwo feels to overcome or defeat the legacy of his lazy father Unoka. Unoka was a lazy man who liked only to play music and drink palm wine, he died with a great deal of debt and no titles and no respect from the tribe. Okonkwo was ashamed of him for as long as he could remember and fought against this throughout his entire life. To overcome the stigma of being Unoka's son he worked incredibly hard to build his fortune, became a great warrior and wrestler, and refused to show any sign of weakness or emotion as he was terribly afraid to show anything like his father's personality.
This in turn led to another conflict which had to do with Okonkwo's temper and the ways that it got him into trouble because he couldn't control it. He was unable to sit back and had to participate in the killing of Ikemefuna, he fired a gun at one of his wives and even beat another wife during peace week, all of which resulted in serious consequences.
Another major conflict has to do with the problems that arose when the missionaries arrived and began teaching the tenets of Christianity and they were in serious conflict with the traditions and beliefs of the tribe. This put the believers in the new faith in direct conflict with the supporters of the traditional practices and ways of worship and led to what some consider the destruction of the Igbo way of life.