Grafting is the mad-scientist way of propagating your plants. Slice a piece from one plant and splice it onto another. Cool! Let's see how.
Many of the plants at your local garden center may be grafted. For example, many hybrid roses are grafted so they grow on the roots of a tougher variety of rose. This helps them stand up to tough conditions and bloom more vigorously. Likewise, most fruit trees are grafted onto smaller variety's roots. This keeps the trees more compact and vigorous. Plants trained as standards, or tree forms, are also usually grafted onto the trunk of another plant.
Somebody really released their inner Dr. Frankenstein when they grafted several varieties on the same tree, so you have an apple tree that produces 'Red Delicious' and 'Golden Delicious' fruits. This is how we got the famous "fruit-cocktail tree."