Hi, Patrick. This is actually a pretty complicated question! Assuming you don't want to know about maturation at all, there are a number of different sorts of mature B cells, among them short-lived plasma cells, long-lived plasma cells, memory b cells.
THough it involves too many process in the course of differentiation, it can be simply taken as, due to the t cell and the t cell mediated responses like cytokine secretion., the B cell arrising from the bone marow in the case of humans differentiates and activates in to either plasma cells and memory cells. Plasma cells secrete antibodies for providing adaptive immunity. Memory cells provide longterm immunity and acts as a memory reservoir for combating against the previously hit diseases. There are also a dseveral steps involved inthe differentiation of the B cells. The best book you can refere is Immunology by Kuby.
There is some very good review about the development of B-cells in swine:
Vet. Res. 37 (2006) 417–441: Development of the neonatal B and T cell repertoire
in swine: implications for comparative and veterinary immunology; John E. BUTLERa*, Marek SINKORAb, Nancy WERTZa, Wolfgang HOLTMEIERc, Caitlin D. LEMKEa