I have to define colony, protectorate, sphere of influence, and economic imperialism (related to imperialism) but can't find any information.
All of these terms that you mention have to do with imperialism. They are different ways in which one country can exert imperial control (or some degree thereof) over another. As you have them listed in the question, the first three go from the one where the imperial power has the most control (colony) to the one where it has the least (spheres of influence).
A colony is a country that is completely controlled by another country. The imperial power actually takes official control over the colony. Some people from the imperial power generally come to live in the colony. Australia was once a colony of Great Britain. India was a colony as well, though many fewer Britons went to India than to Australia.
A protectorate is a country that is not actually administered by the imperial power. It is a sovereign nation, but it is under the protection of the imperial power. The imperial power generally has some control of the foreign affairs of the protectorate. Egypt was once a British protectorate. Small Pacific Ocean countries like the Federated States of Micronesia are US protectorates (though the US has not power over their foreign affairs).
A sphere of influence is a region where the imperial power has no official control, but where it dominates trade. A good example of this has been US domination of trade in the Americas. At various times, the US has so dominated trade with the countries of Latin America that those countries were clearly within the US sphere of influence.
Economic imperialism is a less precise term. It generally refers to a situation in which an economic power uses its economic power to assert control over another country. It does not actually control the weaker country, but it uses its economic might to push that country around. The term also refers to a situation where the stronger country uses the weaker country’s economy for its own benefit (this can also be called neo-colonialism). Economic imperialism is less clearly identifiable as there is no formal relationship between the two countries. Different observers may differ as to when a rich country is exerting economic imperialism over a poorer country (as opposed to just trading with that country).