T cells are named after the organ they develop in, the Thymus.
Cytotoxic T cells (CTLs) are a special type of T cells with the ability to kill other cells. Usually they kill cells that have been infected with a virus. This way the virus can no longer use these cells for replicating inside of them. CTLs secrete cytotoxins to kill the other cells and are therefore called “cytotoxic” T cells.