A classic but not forgotten masterpiece by Sid Meier.
Your first action, I guess just like Columbus, is to name the new world you have just discovered. This world can be a randomly generated or you could choose to play a map identical to the Americas. After naming the new world it is time to found your first colony. Running your colony works somewhat similar to running a city in Civilization (also by Sid Meier) but with a few twists. Just like in Civilization your colony has a few tiles of land surrounding it that you may choose to build a few improvements on and assign colonists to work on.
Differences from Civilization are the elements of religion and liberty. The amount of religion amongst your colonies, which is increased by putting priests to work and building churches, determine the generation of new colonists in your homeland who are looking to migrate to the new world. These can vary from your standard colonists who can do pretty much anything but excels at nothing, to specialists who may be a carpenter, priest, statesman etc. who will increase the productivity of your colony significantly in their relative area of expertise.
The level of liberty increases the work-rate of your colonists. Liberty is increased by building city-halls, printing presses and the like, but be careful as gaining liberty also has its drawbacks as it will also increase the taxes you pay to your home country.
In the new world you set out to build your own little empire and in doing so you deal with the other 3 European powers and the native tribes. How you deal with them is up to you. You can attempt to maintain a peaceful relationship with all different parties or pillage and plunder away but ultimately you will be engaged in a war for independence with your home country.
The entire game builds up to this final battle to gain independence to be free of interference and taxes from "home" and win the game. You start this fight after building up enough liberty and declaring independence but not surprisingly the monarch back home does not sit back and accept this declaration. The epic battle that follows is probably the toughest but most fun part of the game. Depending on how your relationship with the natives and other European powers has been during the game they may choose to help you in your fight for independence or they might come around for some payback when you are vulnerable.
After the smoke has cleared and you are the one left standing your home country accepts your independence and you win the game.