What do the different types of T cells do?

Memory T cells are long-lived T cells that remain after exposure to an antigen and mediate a more rapid and more potent response upon re-exposure to the same antigen. 

There are 5 types of memory T cells. Each has a specialized function. 

1. Central memory T cells (TCM): Central memory T cells are located in lymphoid organs and bone marrow and act as long-term guardians of the immune system. They have a high proliferative potential and provide sustained immune memory by proliferating in the secondary lymphoid organs and producing a supply of new effectors. TCM cells orchestrate coordinated immune responses when re-exposed to specific antigens.  
2. Effector memory T cells (TEM): Effector memory T cells provide an immediate, but not sustained, defense at sites of pathogen entry. TEM cells are present primarily in peripheral tissues such as the skin and mucosa in pre-activated form, which enables them to initiate immediate action on recognizing a pathogen. On encountering pathogens, they immediately secrete cytokines, directly killing infected cells and containing infections at the site of entry. 
3. Tissue-resident memory T cells (TRM): Tissue-resident memory cells remain settled over long periods of time within barrier tissues of specific organs such as the lungs, skin, and intestine, where they have previously encountered pathogens. This enables them to accelerate pathogen clearance and provide immediate immune protection on re-exposure to antigens encountered at body surfaces.  
4. Stem cell-like memory T cells (TSCM): A newly identified subset of memory T cells, TSCM cells contribute to long-term maintenance of immune memory by consistent self-renewing, long life span, apoptosis resistance, and rapid differentiation into effector T cells upon exposure to antigens. These properties enable stem cell-like memory T cells to provide continuous immune protection against previously encountered pathogens.
5. Virtual memory T cells (TVM): Virtual memory T cells are T cells that acquire a memory phenotype despite not encountering specific antigens. Their role in providing protective immunity against pathogens is not well understood although it is thought to be associated with the ability of TVM cells to produce various cytokines.