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B cells are part of the humoral immunity (not the cell-mediated immunity). Leukocytes are any white blood cell (B cells, T cells, macrophages, granulocytes, etc.), only some of which are involved in cell-mediated immunity.
In a humoral immune response, the production of cytokines by T-cells is what actually causes the B cells to have their immune response and start manufacturing antibodies. In laymans terms, the cytokines tell the B cells that something is up, and that they need to start getting ready to defend the body.