Phagocytes are cells found in the bloodstream and protect the body by ingesting and destroying foreign cells. These include, bacteria, dead or dying cells and other invading cells. They are part of the immune system. Chemical signals from invading cells are released and receptors on phagocytes detect these cells which are not "self" and they then proceed to engulf and destroy those cells.
Lymphocytes are part of the immune system. There are three types, T cells, B cells and Natural Killer cells. Natural killer cells are able to recognize and destroy altered cells as in tumor cells or cells that have been infected by viruses. B cells manufacture antibodies which neutralize bacteria and viruses. T cells are of two kinds. Helper T cells produce cytokines that direct the immune response. And other T cells produce granules that cause infected cells to die. Both B and T cells are involved in the production of memory cells. These provide long lasting immunity to the specific pathogen or germ.