Dutch-Portuguese War
The Dutch-Portuguese War (1602-1640) was a conflict resulting from Dutch companies (most of which the Dutch government had a large stake in) invading and impeding on Portuguese land and colonies in the Americas, Africa, India and the Far East. Related to the Eighty Years' War, the conflict served as a means for the Dutch Republic to gain more land with the most prolific ambition to gain a foothold in the New World at the cost of Portuguese settlements. English forces also assisted the Dutch at various points during the war.
Initiated due to Dutch attacked on Portuguese settlements and trade lines, the conflict consisted mainly of skirmishes and battles fought over Portuguese ports, settlements and supplies. The initial conflict was focused in East India, where the Dutch met with great success in their capturing of land and supplies. Following their conquering there they turned their attention to Portuguese settlements in the New World, hoping for repeated success.
Though initially successful in their endeavors, shifting alliances and reinforcement lines eventually allowed the Portuguese to repel the Dutch in Brazil. Similarly, the Dutch were repelled from Africa where they met with equally little success overall. With their interests at stake the Dutch eventually looked to bargain for peace in order to retain whatever possible in Brazil and Africa.
However, it was a watershed mark that indicated the beginnings of the decline of the first European Colonial Empire, as the strains of the war bled the shipping and trade Portugal so depended on and saw an entire quarter of their Empire in Asia fall away, marking the true birth of a Dutch Empire.
The Other Wiki offers a comprehensive article on the three major campaigns of the war.