The fundamental conflict explored by Achebe in Things Fall Apart is that of tradition versus change. The changes that take place among the Ibo people after the arrival of Christian missionaries (and during Okonkwo's exile) are profound. Many people not previously held in high esteem by the Igbo are converting to Christianity and gaining prestige as a result of their actions. When Okonkwo returns from his exile, he finds that the religion has destroyed much of the coherence of Ibo society. This trend is exacerbated by the imposition of a colonial society, complete with an exploitative palm oil industry. Even his son has joined the ranks of the converted. Okonkwo tries but fails to marshal support for a return to traditional life, and so he kills himself, having become an anachronism in the span of a single life. He is in many ways a casualty of this basic conflict.
As the other answers to this question suggest, the major conflict in Things Fall Apart is between the native Igbo culture and that of white colonial culture. Indeed, the book ends with white colonists gaining greater and greater control over the native populations, and so much of Chinua Achebe's novel focuses on the conflict inherent in colonial expansion. The most dramatic expression of the conflict comes at the end, at which point Okonkwo succumbs to the despair of losing his traditional culture and kills himself.
However, there is another conflict at work here, the conflict between patriarchal aggression and the community. Fearing failure and weakness above all else, Okonkwo leads a strict life based on physical prowess and domination. Okonkwo views any kind of failure as weak and detestable, and so he becomes an image of the radically individualistic and proud patriarch. This aggressive pride conflicts with the communal focus and culture of Umuofia, and it ultimately leads to Okonkwo's exile and isolation. Thus, one of the book's main conflicts illustrates the damaging effect of patriarchal aggression and shows the ways in which unchecked, destructive male anger stands in opposition to the harmony of the community.
There are really three separate major conflicts in Things Fall Apart by Chinua Achebe, one of which is individual and two of which are cultural.
On the individual level, the major conflict is that of the Igbo warrior Okonkwo in his quest to achieve greatness. His major struggle is against his own "chi" or divine nature, which does not have a great destiny. Thus, in many ways, Okonkwo is fighting against his own nature and fate, although this conflict is also externalized in the plot of the novel. He angers the earth-goddess Ani and the religious authorities and traditions of his village. After he returns from exile, he comes into conflict with the British.
On a cultural level, there are two major struggles in the book. The first is between the Igbo and the colonial British and the second is between traditional Igbo religion and Christianity. The characters of the novel are often caught between these opposing forces.