Plant surgery: The most challenging form of propagation
The basics: Most grafting takes place between members of the same species, although sometimes you can get away with grafting in the same genus (ex. peaches on a plum tree). The bottom (stock, or rootstock) and top (scion) meet at a point called, appropriately, the "union." There are many, many different types of grafting, among them veneer grafting and chip budding.
The not-so-basics: The term "surgical" is usually somewhere near whenever "grafting" is mentioned, and with good reason. This is a difficult propagation method, requiring proper timing, tools, and expertise - three very good reasons why grafted plants are usually so expensive in garden centers. Still, it isn't brain surgery, and a good reference book should be sufficient for one willing to take on the challenge.