Phosphorylation is the addition of phosphate groups (PO4-) to molecules. Though it was known to be important in the normal metabolic pathways that keeps us alive, the Nobel prize winning discovery (by Edmond H. Fischer and Edwin G. Krebs) was in its use in the regulation of biochemical pathways. Turns out that much of the ability of cells to respond to different stimuli depend on cascades of phosphorylation events that modify the function of downstream enzymes. For example, a DNA-damage sensing protein could respond to damage by phosphorylating protein X, which upon phosphorylation phosphorylates protein Y, which in turn, phosphorylates protein Z. Protein Z, in turn, activates genes important in DNA-repair. This makes sense, if you imagine there are different proteins that detect different kinds of DNA-damage, and they all need to turn on the same gene. Also, DNA-damage can require the activation of a number of genes at different times. This can all be handled by changing aspects of the phosphorylation cascade. Replace DNA-damage with everything that a cell can do, and you have an idea of how cells work.
Due to reasons that are probably historical, enzymes that do phosphorylation are called kinases, while enzymes that remove phosphate groups are called phosphatases.
The kinds of phosphorylation important in signal transduction are tyrosine phosphorylation and serine/threonine phosphorylation on proteins.