This will be a long and boring answer, but it's a good question that deserves a full answer
A beauty dish is a large round reflector that has a deflector fitted to prevent a hotspot of light from the flashtube directly hitting the subject. It has a parabolic shape, which controls the spread of light extremely well.
Beauty dishes are really designed for photographing beautiful models who have very high cheekbones and perfect skin, so basically east Asian and African people, although some European people were born lucky too
Used with the wrong model, they should be called "Ugly dish" instead of "Beauty dish" But they're also great for photographing people with very "craggy" faces, where we want to accentuate features, scars and so on, and I've often used them for dramatic product photography too.
Silver ones create a more dramatic lighting effect than white ones but they also amplify any faults in either their use or in the model's complexion.
The leading make is Mola, there is simply nothing like them in either efficiency or cost. See
https://www.mola-light.com/product/demi/
Very few people can afford Mola and there used to be a lot of copies around that worked well, a price compromise that suited most people. But, times change, the metal ones are large, heavy and can't be taken apart, so everyone started wanting folding ones made from fabric - not as good but more convenient. Most were very badly designed, so I became involved in designing a very good folding one with the best possible shape, depth, deflector and materials. This could be used both as a beauty dish and as a softbox, just remove the deflector and add either one or two of the supplied diffusers and we end up with a softbox. This, inevitably, was then copied by the myriad of largely backstreet Chinese factories, which as always found ways of simplifying production and cutting down on material costs, and sold by other firms all over the world. I haven't seen/tried most of them, but those I have tried are nothing like a real beauty dish, the shapes seem to me to be wrong and the materials seem to me to be flimsy, but they can still be a useful tool, so there's nothing wrong with buying one if you want to.
