B-cells are the cells in a vertebrate immune system that makes antibodies.
B-cells make antibodies by first randomly rearranging DNA that encode the two proteins that make up antibodies. Only the ones that make a functional antibody survive. B-cells that end up making antibodies to the animal itself are quickly killed.
At first, B-cells have cell-bound antibodies. When a B-cell encounters an antigen that the antibody recognizes, the B-cell (with the help of T-cells), begin to multiply, and mature. One of the ways in which B-cells mature results in plasma cells, which hang out in the bloodstream pumping out their antibodies.
The naming and discovery of B-cells is a fascinating story.
People knew that animals made antibodies, but for a long time did not know where and how they were made. One day, a graduate student was conducting an undergraduate course on immunology, and the demonstration for the class was the production of antibodies in chickens. Typically, this involves injecting chickens with an antigen, and collecting the chicken blood sometime afterwards, and demonstrating that there is an antibody that recognizes the antigen in only the chickens that were injected.
One day, this demonstration, which had worked every single time in the past, did not work. The graduate student, upon investigation, found that the chickens used in the latest demonstration had been used by another graduate student, who was investigating the fairly unpopular avian organ, the Bursa of Fabricius. In fact, this other graduate student had been cutting the Bursa out of the chickens, but in the interests of cost-savings, the chickens were used in the undergraduate class.
So upon further investigation, they discovered that a cell in the Bursa, which they named the B-cell (for Bursa cell) was the cell that was making antibodies in chickens. This landmark discovery was published in the journal Poultry Science. (I kid you not.)
Eventually, the importance of this discovery was understood. And in a very lucky break for all immunologists (and undergraduates studying immunology), the cells in mammals that make antibodies just happened to come from the bone marrow, so the B-cell can also stand for Bone marrow-cells. | 
