Imperialism: A policy of extending a country's power and influence through colonization, use of military force, or other means.
Neo-imperialism: Domination and sometimes even hegemony over others primarily by way of formally free legal agreements, economic power, and cultural influence.
Historically, imperialism was in the form of colonization. A powerful country would attack another country, militarily occupy it, and extract the country's resources.
In modern day, imperialism works more discretely. Instead of taking a country outright, powerful countries such as USA will try to ensure other countries work in favor of America through establishing puppet states. These states are nominally independent but in practice controlled by USA through a proxy.