Hard wheels: rapid removal of metal or rust, good for profiling, make a mess. They can put a fair bit of heat into a workpiece. Cheap. Don't bear too hard. They won't cut any faster. ou can excavate with the edge.
Cutoff wheels: For, believe it or not, cutting. Also useful for grooving material. Come in a variety of thicknesses, 0.045 and 0.060 being real common. They don't take a lot of side force, an throw chunks when they break, The chunks are sharp and moving fast and will penetrate flesh. Use a gaurd. Keep these in line with the cut, as ANY significant side load will cause failure. Material up to about 1" diameter: just cut through it. Cutting sheet it is best to take several passes, grooving deeper each time, as trying to do this in one shot will allow the edge of the material to disintegrate the wheel. If the material (or wheel) gets too hot, the wheel will disintegrate.
'Elastic' cutoff/grinding wheels: A variety of manufacturers make wheels that are usable for cutoff or grinding Pferd is that brand I like. They take side load better than regular cutoff wheels, but arn't great for grinding and will break. They are my preference for cutoff work unless I need a narrow kerf, as they are less likely to shatter. Available from about 0.060" to 0.250" thck, the thinner ones being intended primarily for cutoff work.
Flap wheels (Tiger wheels): The cat's @ss for smoothing and polishing. 40grit for reasonable metal removal, up to 120 grit for not-quite-mirror smooth. decent life, but not real high removal rates. Takes practice to avoid making a noticeably wavy surface, as they are soft. Good genera purpose cleaning and light polishing. Not very heat tolerant in most makes.
Wire wheels: Great for slag removal, paint removal, derusting, preweld prep (except soft metals like aluminum, where they tend to bury the crud in the surface). I always keep one in hand when stick welding. Don't force them. Lighter pressure and more passes in more effective than heavy passes. Vary the direction. They throw wires out at 8 billion miles per hour, so safety glasses and face shield, and no exposed skin. Leather recommended. Keep the wires perpendicular to the surface or they fly out more. I prefer the radial type to the cup type for most things.
I tend to keep a couple different setups handy, a wire wheel and hard wheel always, and a flap wheel usually.
I may not be good looking, but I make up for it with my dazzling lack of personality