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There has been street photography ever since the camera was first invented in the 1830s. But if you look at those early examples what you mostly see is just that - streets. Emulsions and lenses were so slow and such long exposures were required that people and traffic mostly disappeared, except for an occasion blur. So you see decades of photos of buildings and avenues with nobody in them.
Once cameras became more portable and film and lenses became faster photographers started trying to do candid photos of life on the street. Photographs like this were documentary in nature but a particular kind. Street photographers have often shot people in candid, unguarded moments. Or captured subjects who knew they were being photographed but still reacting spontaneously to the camera.
A great example of early street photography was Eugene Atget, who began capturing images of what he called "old Paris" at the very end of the 19th century. His photos of cityscapes and the people of Paris are some of the greatest ever. There have been dozens of really fine street photographers over the history of photography but perhaps the best known and regarded was Henri Cartier-Bresson. He was known for the term "the decisive moment" and in many ways has set the standard for what street photography should be. Cartier-Bresson shot BW photos with a Leica at a time when 35mm photography was first being established. With a small camera he could remain unobtrusive and photograph his subjects without disturbing or altering their normal behavior. In a world used to large view cameras, even when he shot subjects aware of what he was doing they tended not to react much or be intimidated by this quiet man with the (relatively) tiny little camera.
Cartier-Bresson almost always shot with a 50mm lens and tried to compose so the final print would not be cropped. His goal was to compose and shoot a photograph just at the precise instant that revealed the truth of whatever was occurring in front of his lens. As I said, this "purest" approach represent what, even to this day, is thought of as the ideal way to do street photos.\
But there really are no rules. Many street photographers nowadays shoot in color. And they use all sorts of cameras and lenses. Some use long lenses and shoot from a long way off so the people in the shots have no idea they are being photographed. Others use cameras like a GoPro or something similar letting them shoot wide-angle, very close up photos without making it obvious what they are doing.
Nowadays, people are much more conscious of being photographed than in the times of Atget or Cartier-Bresson. Given the anxieties of the age, a lot of people are reluctant to have their photos taken by strangers. One approach is not to attempt candid photos at all. Bring along some examples of your work and when you see people out and about you'd like to photograph just approach them openly, in a friendly manner, tell them you'd like to shoot them and show them your work. Some will refuse, so just smile and move on. Most will agree if you approach them in a pleasant enough manner.
A friendly, non-threating personality goes a long way in getting strangers to trust you to shoot photos of them.
By the way, you can't legally use street photos like this for commercial purposes without a commercial release. Photographers who do want commercial rights usually carry examples of their work, even a brochure or booklet of their photos, and something that describes what use of the photos subjects are agreeing to if they sign a release. Photographers can agree to provide copies of the photos as compensation or payment in the future if the image is used to create income. There are quite a number of instructional resources online on the subject if you want to go into this further.