Most of the countries of Europe established colonies across the globe from as early as the Renaissance. The discovery, claiming and use of the colony for its own interests really grew out of the Industrial Revolution in the 1700s and 1800s, where industry in European countries was demanding more resources to allow more industrial growth to occur.
The British, with one of the best navies in support, were one of the largest colonizers of non-European countries. Coupled with a need for resources was a belief that British represented the highest level of humanity and that those colonized countries were simply there to be taken over by a smarter superior race. British Colonies were set up in Canada, India, the West Indies (Caribbean Islands), South Africa, Rhodesia (now Zimbabwe), India, Burma, Malaysia, new Zealand and Australia. There were many other smaller nations too in these areas that made up the British Empire.
France colonized parts of the United States (Louisiana) as well as Canada (Quebec), the Ivory Coast, Senegal and many other African nations as well as Vietnam (whose eviction of the French caused the USA to intervene and commence a track towards the Vietnam War).
Spain was another country that colonized several countries. Central America was colonized by the Spanish, including parts of California and Mexico as well as countries such as Costa Rica, El Salvador, Puerto rico and Cuba. Spain also colonized many countries of South America including Argentina, Chile, Colombia, Bolivia, Ecuador, Paraguay and Uruguay.
Portugal colonized Brazil, Mozambique (and other smaller African nations), as well as Timor to the North of Australia.
Holland, Sweden, Denmark also held smaller colonies but it is fair to say that Britain, France, Spain and Portugal were the largest colonizers of Europe in the 1700s and 1800s.