In an immediate sense Columbus was not important. He came, went, came a couple more times, and then left again. What matters though is Columbus finally proved to Europeans that there was something out there.
Europe was slightly ahead of the Islamic world and slightly behind the Chinese in the sense of scientific accomplishments in 1491. Access to the resources and opportunities in the New World allowed Europeans to advance well past others. For those all ready in the Americas things changed as well. The Spaniards and others brought diseases which the Indians had no immunity to, the weapons and tactics of the newcomers overpowered those who resisted, and the regional empires crumbled. The vacuum was quickly filled by the newcomers.
There were several approaches to colonization. The Spaniards mostly came for "Gold, God, and Glory." This led to an overwhelming male migration to the New Spain. Without the oppurtunity to marry Spanish women many turned to marrying Indians. The Mestizo race was born in the early days. Indian elements transfused with Spanish culture creating something new. The Our Lady of Guadalupe embodies this revolution: Spanish culture with an American identity. Military conflicts with Indians died off quickly besides a few areas of non-intermixing in Chile, Argentina, and Mexico.
The English had the second approach. They sought to transplant their identity to the Americas. They brought women with them along with various creeds of Christianity. The English saw themselves as completely separate from the Indians. Wars would rage between the cultural descendants of the English and the Indians for centuries. However, the English ideals of freedom and liberty (which was not perfectly practiced) allowed for the United States to become realized. The United States has sense experienced triumph and tragedy; and has become a become of freedom for the rest of the world.
Others would come as well. The French had limited success creating a New France (Quebec) but on the frontier they copied the Spanish model of men mixing in with Indians. Catholic tribes and better trading relations was common wherever the French frontier was. The Dutch came and went quickly from North America but left a legacy of arming Indians against other European powers. The colony wars were much bloodier because of them. The Portuguese have Brazil as their landmark. For a while Brazil was European in the sense of its culture. It even acted as the seat of government for Portugal-in-exile.
And let us not forget the Africans who were sold into slavery by other tribes and Arabs to Europeans who saw them as draft animals. Those of African descent have struggled for freedom and reached various results. Jim Crow laws stunted political growth for 100 years in the United States while those in the Caribbean gained control via the democratic process. After the end of slavery in Brazil a new culture was born as the races began to mix together.
Christopher Columbus was not the first one to discover America but he was the start of a revolution. With him the geography of the New World was changed forever and the earth we live on along with it.