All whisk(e)y requires two things for production: grain and a cask, usually of oak. You make a beer (or mash), run it through a still, and put the clear stuff that comes out into a barrel. As the spirit sits in the barrel, "hotter" alcohols evaporate away, flavors and wood sugars are extracted from the oak, and oxidation occurs, softening and smoothing out the spirit over time.
The vast differences in flavor occur from the grain bill (or recipe of types/amounts of grain used), the barrels ― wood type, size, toast, and amount of use ― yeast, aging process, and to some degree geography. All these factors contribute to the diversity of styles and flavors. The category details and variation are unfortunately too enormous to detail here, but suffice to say that there exists a large swath of flavors from dominating to subtle, grainlike to smoky to woody. There's something for everybody.
The main categories, at a glance:
Scotch whisky: The default when you think of whisky. Flavors range from peaty/smoky to saline, medicinal, leathery. Usually made from mostly barley, aged in used barrels.
Irish whiskey: Usually lighter in flavor than Scotch and often made with a blend of grains, aged in used barrels.
American whiskey: Corn usually dominates the grain bill in bourbon, rye, and Tennessee styles, and often goes into new American oak barrels for big nutty and vanilla flavors.
Japanese whisky: No longer an oddity, since Japan's output been so consistently solid. A little lighter and more floral than Scotch, but made with traditional Scottish methods, often making it indistinguishable.
Single-malt whiskey: Any whisky that's distilled, aged, and bottled by a single distillery and made entirely of malted grain, typically barley. A robust, full-flavored spirit.
Canadian whisky: A blended whisky (see below) originally made with mostly
rye, but primarily corn-based. Notable for its popularity during Prohibition.
Blended whisky: A whisky that combines several single malts along with an unaged neutral grain spirit to produce a lighter spirit.
White Dog: Typically an unaged American whiskey that's come a long way from moonshine. Tends to have more of the grain flavor.