The cucumber (Cucumis sativus) is the fruit of a creeping vine native to India, where it has been grown since around 1500 BCE and brought to the Mediterranean around a thousand years later. The cucumber is a fruit - it's in the same family as the squash and in the same genus as the muskmelon.
There are five main varieties of cucumber, which are broadly broken down into slicing types and pickling types. Pickling cucumbers are small and warty, with thin skins that are pale green to green in color. They are typically pickled, either in brine or through a lactic acid fermentation. Smaller versions of these cucumbers are commonly called gherkins, which are actually a related species -- Cucumis anguria. In France, tiny gherkins called cornichons are typically served with pate or charcuterie. Slicing cucumbers, like the typical supermarket cucumber (which is waxed) or the often plastic-wrapped English cucumber (which is not) are typically eaten raw, especially in salads.
Cucumbers have a high water content and a central column of seeds, which are immature and edible. The aroma of cucumber develops when it is chewed or cut, and has a faintly melon-y, watery aldehydic aroma that is cool and clean. The peel of most cucumbers has a high concentration of bitter chemicals called cucurbitacins, which discourage pests.
In cocktail usage, cucumbers offer a refreshing aroma and flavor to drinks. Cucumber is often paired with gin (and in fact, some gins are flavored with cucumber), but interestingly enough, violet pairs very well with cucumber.