Which of the two types of immunity require white blood cells?

 Immunity always needs white blood cells, which can be granulocytes ( neutrophils, basophils, monocytes, macrophages and oesinophils ) , or lymphocytes ( which can be B cells or T cells ) 

There are two systems : Innate and Adaptive immune response 

There are 2 types of immunity : humoral and cellular within the Adaptive immune response 

Humoral immunity refers to B lymphocytes. When there is a pathogen circulating in the peripheral blood flow, B lymphocytes recognises their surface antigens ( through the aid of antigen presenting cells like macrophages and dendritic cells) and induce production of plasma cells, which faciliate production of immunoglobulins ( 5 types), which aid to the removal of pathogens ( but not killing them) 

Once the pathogens get into tissue or cells, humoral immunity can not be effective anymore. So this is when cellular ( or cell-mediated immunity ) is mobilised. These are T lymphocytes. There are a few types. T CD4 ( or T helper cells), which act as co-stimulation for B cells production of immunoglobulins and fight against certain pathogens. T CD8 cells ( cytotoxic T cells ) are the ones that have major role in killing all cells and tissues that are infected by secreting perforins which basically create pores in the infected cells, leading to cell death ( death by apoptosis). There are also NK cells ( natural killer cells ) which basically are licensed to kill anything that is " not normal " such as malignant cells. 

The B and T cells response above belong to the adaptive-immune response. 

Another immune reponse system which is short-lived and immediate is the INNATE immune response which include the granulocytes stated at the beginning to this answer. Upon infection, these cells will be immobilised immediately to rush to the site of infection or exposure to seek and destroy any thing that is considered foreign ( neutrophils will fight against bacteria, oesinophils and basophils fight against parasitic infection and induce allergic and hypersensitivy responses ). Note that the innate immune response is for quick response, and they are not involved in long-term protection as is provided by the adaptive-immune response. They are also not likely to be able to fight against viral infections.