Jazz shoes are the type of shoes which are worn while dancing Jazz. If you are wondering why a separate type of shoes is needed for dancing, the answer is simple. For any type of activity you need gear which allow you to do that act better. For example, for swimming you need swim suits which allow you to swim better than usual clothing (an understatement but you get the idea). Similarly dance shoes are needed for your feet to perform better while dancing. Normal street shoes do not provide the flexibility or comfort which dance shoes offer. And among dance shoes, Jazz shoes are probably the best in terms of flexibility and comfort.
Jazz shoes are of course not the only type are available. You have ballet slippers, pointe shoes, tap shoes, and dance sneakers. Ballet shoes (slippers and pointe) and tap shoes are specially made for these dance forms. Ballet, especially pointe shoes, make it easier for ballet dancers to go en pointe that is rising on their toes’ points. They are also very light weight which makes performing many of the complicated ballet steps easier. Tap shoes have special plates attached to them which add to the sounds produced while tap dancing (the dance being as much about sounds as movements). Tap shoes are always high heeled.
Dance sneakers are used for modern dancing and have very rigid upper parts which do make them good for many modern dance styles such as street dances. They typically have split soles. Jazz shoes, generally made from supple leather or from canvas (and hence have more snug uppers), provide more comfort and flexibility when compared to dance sneakers and are mostly used while dancing jazz, hip hop, and other modern dance forms which require a decent amount of flexibility, speed, and support. Also, these shoes are slippery which helps you slide on dance floors easily.
Jazz shoes have leather soles which could be either full or split (some use rubber). The split soled shoes provide more flexibility. Traditional shoes of this type come with laces though these days you also get the slip on variety. They are either tan or black, sometimes embellished with designs for the purpose of the dance. There are many different types of Jazz dancing shoes. You have Jazz boots, Jazz sandals, Jazz sneakers, and so on. Some of them come with suede patches which provide extra flexibility. Others come with neoprene insert which helps the arch. Yet others come with aerated arches which provide better breathing. All in all, Jazz shoes provide quite a lot of variety.