Imperialism is simply a process in which some countries dominate other countries. They take those countries (sometimes, but not always, actually conquering them and ruling over them) and impose an unequal relationship on them. In this relationship, the periphery (the dominated country) is made to serve the core (the imperial power).
The term "new imperialism" is used to refer to a wave of mostly European imperialism that happened in the late 1800s and early 1900s. This wave is said to have been slightly different than the previous period of imperialism in which European countries created empires in the Americas.
For example, the first period had more colonialism (where many people from the "mother country" went to live in a colony) while the new imperialism had more imperialism (where only a few of the Europeans would actually go live in and rule the dominated country). The new imperialism is said to have been motivated more by a desire for economic benefits (new markets and sources of raw material) than the old. It is also said to have been motivated more by a desire to "civilize" and Christianize the rest of the world.