The banana is the fruit (actually a berry) of a large flowering plant in the genus Musa. This family covers a wide range of species, of which many have been crossed with each other to produce the common supermarket banana. While they are in the same genus, bananas are usually divided into sweet, dessert bananas and starchy, firmer species, which are commonly called plantains.
The banana has been cultivated since ancient times, and was probably domesticated first in Papua New Guinea between 5,000 and 8,000 BCE. From there they spread to Africa in the first millenium BCE, and were brought to introduced into the Middle East by 650 CE. In the 16th century, the banana was brought to South America by Portugese explorers, but were uncommon in North America and Europe into the 19th century.
Most of the bananas in supermarkets in the United States are from the Cavendish group of cultivars, though the spread of a fungus called Panama TR4 will ultimately remove the Cavendish from mass production.
Interestingly enough, bananas are slightly radioactive (they contain high levels of potassium, some of which is Potassium-40), and when they're ripe, are fluorescent. The aroma of ripe bananas is mainly from isoamyl acetate. Bananas are picked while green, and shipped cold, then held in a room full of ethylene gas, which not only induces the production of amylase, which converts starches to sugars and changes the color of the skin, but also pectinase, which softens the fruit.
Bananas should be stored at room temperature, and if additional ripening is desired, in a paper bag with an apple or tomato (both of which release ethylene). Refrigerating bananas will blacken the skin, but will not affect the quality of the fruit. Bananas are ripe with intensely fragrant, soft, and the skin has started to be marked with black spots.
Strangely enough, bananas are low in fat, but seem fatty because of their slippery skin and creamy texture. They pose problems in cocktails -- muddling bananas into a drink leaves the texture strangely slippery, but fat washing them in alcohol does not work.