Answer:
T cells and B cells are both types of lymphocyte, the major part of our ADAPTIVE IMMUNE SYSTEM. By this, we mean that these cells are unique to every single person, and no two people will ever have the same array of B and T Cells.
Both B and T cells begin life in the bone marrow (that what the B stands for). T cells when they are very young and immature then get into the blood and travel to the thymus gland near the heart (that's what the T stands for). Here T cells become either T-helper cells (Th) or T-killer cells (CTL's) and everybody then travels to the lymph nodes.
In very general terms, this is how it all works -
When we get infected by an invader (say, a bacteria on a splinter in your finger), massive bacteria-eating cells (called macrophages) eat them, and spit out what's left. These bits of bacteria are then eaten by very special cells called dendritic cells (which look a bit like starfish). These then travel out of the tissues of your finger and into nearby lymph nodes. Here they show these bits of bacteria to T cells. Now T cells are really special - they don't just recognise any bacteria, they only recognise 1. So, through lots of complex processes, the dendritic cell eventually finds the T cell that recognises the bacteria that it has onboard. When this happens, the T cell multiplies massively in the lymph node.
The T cell then travels directly to the site of infection (e.g. your finger) and secretes cytokines (these are chemicals that tell the other cells of the immune system what to do).
B cells are quite different, even though under a microscope they look very like their T-cell brothers. The B-cells main job, once activated, is to sit in the bone marrow and churn out tonnes of antibodies. These antibodies travel back to your finger, bind onto the bacteria that are still there, and make it easier for the big macrophages to eat them.
This way, your immune can produce a very specific response to pretty much any infection it comes into contact with.