Our map only covers Europe. We represent the European colonization of distant lands with Colonial projects. To build many colonies, you first have to establish (i.e. build) a trading company. Your choices are the East or West India companies, and they unlock different colonies. Atlantic Access (a resource on the map) doubles the construction speed of all colonies, and Slaves (available from African colonies) increase the construction speed of others still further. If you're worried about another civ building your colony, it may we worthwhile to pillage the "Colonial Trade Route" which provides them with Atlantic Access.
Colonies are required for a number of UHVs, and also provide access to many luxury resources not found in Europe (or only in a few spots). Building them will allow your cities to grow large and powerful.