What was the cause of the French and Indian war?

The French and Indian War started because the British settlers were going farther into Indian lands in the Ohio Valley, and the French were also active in the region, mostly as trappers. The Indians fought on both sides, but they were mainly allies of the French.

Wars Between the British and French
The war was for the control of the rich Ohio Valley and basically was a part of the larger Seven Years' War in Europe. The local name was the French and Indian War and from some perspectives, the outbreak of actual hostilities began in the American colonies of England and France.

Although struggles for supremacy had been going on for many decades between France and England in the New World, hostilities intensified in the early 1750's as both English and French settlers had attempted to colonize land in the Ohio River Valley, near present day Pittsburgh, Pennsylvania. The English settlers, who had moved northwest from Virginia, and French settlers, who had moved east from the Great Lakes, or south from Canada, each thought they owned the rights to the land.

Basically, because of European competition, both France and England wanted to extend their "world empires" into the Ohio River Valley. And each recruited Indian tribes (the original inhabitants) to fight on their side.

(see related question)
The French and the British both fought for land, and Indians were caught in the middle. The Indians defended themselves. This war lasted seven years. A major surprise was that this battle was fought hand to hand. The French and British were used to honorable fighting styles, in which the opposing force could see the other from a mile away. The Indians would hide in trees and bushes until they had the tactical advantage.