There are three or four main areas of scooter manufacture, Japan, China, Europe (Italy) and Taiwan.
In Japan the "big three" are Suzuki, Honda and Yamaha. Kawasaki don't make scooters! The Japanese scooters tend to be big and expensive (though Honda has some small models). They have the quality and reliability that the Japanese auto and motorcycle industry is famous for and they are sold through dealerships in the US, with good aftermarket service.
In Europe most scooters are made in Italy by manufacturers such as Aprila, Piaggip, Garelli, Gilera, and Vespa (though a number of these brands are actually subsiduaries of Piaggio). There are a few other European scooter makers like Peugeot (France) and Sachs (Germany), but the Italian scooters dominate the market. The quality of European scooters is generally very good and they often have a little more "design flair" than Japanese scooters and many more 50cc, 150cc and 250cc models are available than from Japan.
In Taiwan Kymco is probably the best known brand. Scooter quality is generally pretty good, approaching that of Japanese and Italian scooters, if perhaps not quite equaling it.
China probably makes more scooters than all other countries combined. It's very difficult to list all the manufacturers and models because there are so many factories making so many essentially identical scooters based on essentially identical parts (often clones of earlier japanese engines and transmissions). Quality control is generally pretty poor. You can get a good one, or you can get a bad one. They are generally sold by importers rather than dealers. The importers simply ship the crated scooter from China to the US (often California), then ship it on to you without even opening the box. What you get is what was packed in China. Some assembly may be required (and usually is). The one major selling point of Chinese scooters is that their price is 1/3 to 1/4 of that of a similar specification Japanese or Italian scooter. Some manufacturers (or at least brand names) are Roketa, Tank, Diamo, Jonway, Jmstar, Lance and Motofino (sounds Italian, but isn't!). "Dealer" support is generally poor, many dealers simply being importers. Finding someone to work on a defective scooter may not be easy.
If you are looking for a US manufacturer of scooters, good luck. There are a few "US" brands, but they mostly contract with chinese scooter makers and the scooters are made and assembled in China (e.g. Lance and Flyscooters). However Vectrix actually make scooters in the US, but they are electric scooters. They look nice and are excellent quality, but have a limited range (about 65 miles if you are lucky, as little as 25 miles in congested traffic) before requiring a 3 hour recharge and the cost is high (around $11, 850). A new set of batteries is around $3000, so although they are very "green", they aren't a low cost mode of transport to buy and they are not really useful for any sort of extended distance travel, though while you still have battery power they can reach 60 mph. There have been rumors that Harley Davidson is thinking of making a scooter, but it would no doubt be one of the larger "luxury" touring scooters if they did, and I doubt it would be inexpensive. In 1960 HD actually did make the 165cc 2-stroke "Topper" scooter, but the model didn't last long.