Cold atmospheric plasma (CAP), an ionized gas operating at room temperature, has been increasingly studied with respect to its potential use in medicine, where its beneficial effects on tumor reduction in oncology have been demonstrated. This review discusses the cellular changes appearing in cell membranes, cytoplasm, various organelles, and DNA content upon cells’ direct or indirect exposure to CAP or CAP-activated media/solutions (PAM), respectively. In addition, the CAP/PAM impact on the main cellular processes of proliferation, migration, protein degradation and various forms of cell death is addressed, especially in light of CAP use in the oncology field of plasma medicine.