Style is the outcome of two major choices made by the photographer.
I've been shooting photography - and street photography - for several years and I still have no idea what my "style" is. More on point, there's no one generic or specific name for it: my style, therefore, would only be defined by description. I can tell my images apart from other photographers, though I'm not sure my audience would posses that sense of distinction. And yet, the more I shoot and improve as a photographer, the less I really care about style or how my work compares/contrasts with the rest of the masses'. I've concluded that there really is no rush for a photographer to define his or her own style. On the other hand, said photographer should be thinking about two elements on a daily basis to improve specifically, to grow and to mature over time, by making definitive choices in subject selection and post processing.
Subject Selection & Post Processing
I believe photographers are usually thinking about these two elements more or less constantly, even if subconsciously. And judging by the billions of images out there, photographers may place more emphasis on either one or both. Woe to the photographer who cares for neither!
A few notes on what I mean by each element.
Subject selection is not only defined by the 'what', but also by the 'how'. I'll use Bruce Gilden as an example here to illustrate the what and the how, since his style is very noticeable and well-known.
What Gilden is notorious for is his close-up photography work of people on the streets using flash. There you have the 'what' and the 'how' for subject selection. If you look through his photography, you may come to the conclusion that his post processing is probably over all receiving a light treatment. His work, an in-your-face style if there ever was one - and quite literally so! - speaks for itself.
It follows that the 'how' is closely related to what gear the photographer is using. Camera choice. Lens choice. Accessories choices. Anything which ultimately impacts the look of an image falls under the Camp of How.
The Camp of How is additionally home to major photography concepts like composition and lighting, to name only two of dozens. While there may be some cross-over here with post processing (cropping and dodging and burning, etc.) the photographer's first choice should be to try to get it right in-camera whenever possible. This means learning the rules so that your images will stand out - or so that they may be broken later for artistic or practical reasons. The point is to know the difference, and this can only be accomplished through an educational process, a learning process, however formal or informal in scope. In short, learn your chops. Style will follow.
Subject selection in itself is self-explanatory but there are subtleties here worth discussing too.
Distance-to-subject is something that beginners and amateurs struggle with, especially in street photography. It boils down to anxiety or a fear of getting close to strangers to take their pictures. But this is one consideration to be aware of, in any genre of photography. This isn't to say that getting close is the imperative but usually there is more impact in an image. On the other hand, the background or environment is equally or more important to frame and distance-to-subject on a larger scale can be perfectly acceptable. The point here is to be aware of what range your subjects are framed.
Perspective is another one of many sub-elements to keep in mind. A common perspective - and mistake - for beginners in street photography is to always shoot from eye level, thus producing a predictable angle from photographer to subject. I've seen a few of my clients and a lot of photographers on the streets never get out of what I call tourist mode. It's as though there's an invisible harness around the photographer's neck preventing him from bending or positioning his camera in such a way to avoid the same boring perspective shots again and again. This is one thing any photographer can change immediately to further develop one's photography and in time, one's style. Get out of tourist mode and change-up the perspective of your images.
The discussion here can be as wide-ranging as any element in subject selection. For me personally, I place as much importance in post processing as I do in composition and lighting. This doesn't mean that I'm intentionally heavy-handed in my post processing; it means I believe that without post-processing, there is less scope for authentic photography, less opportunity to express individuality, i.e., style.
On the extreme end to my position are those who claim to not use any post processing whatsoever. They are the "straight out of camera" camp. Obviously, I'm at odds with this philosophy, especially if we're talking about street photography versus those shooting in environments (like studio) where lighting and other elements are controllable.
In my view, if you choose not to edit then you're choosing to let your gear dictate your post look as much as you're allowing for your subject selection inputs. And if you're shooting in RAW, you always need some light editing to bring about the best in your image. If you aren't, then you're simply not making use of your file's inherent capabilities.
Which brings up the matter of RAW versus jpeg. If you have the choice, you should be shooting in RAW. Jpeg is not what any serious photographer should be considering apart from a few exceptions. I won't be discussing those exceptions here for the sake of being on-topic and for brevity. But here's a video series explaining why you should be shooting in RAW.
Global Adjustments vs. Local Adjustments
Photographers should be thinking in terms of global versus local adjustments when it comes to post processing time. Developing a sense of when to use each will not only improve your images over time but will also lead to a noticeable change in how your images look. This is definitely Style Country here, where one or multiple changes could dramatically change appearance.
Global Adjustments are broad, sweeping changes which affect the entire image, like a black and white conversion. Filters and plugins of various kinds also affect images globally, as do most of the sliders in photo editing software.
Local adjustments will only affect targeted areas of an image. There is a large set of tools and procedures for this in programs like Photoshop, like radial and graduated filters, dodging and burning, spot healing and patch tools, adjustment brushes, etc. In many instances the photographer is attempting to make his subject stand out from the background, and local adjustments are fit for this kind of procedure.
The tendency for beginners is to only use global adjustments in post-processing. In my view, there's a new world awaiting those who haven't explored local adjustments. Additionally, I believe it's only by learning and making local adjustments that will ultimately make your photography stand apart, or at least add to and enhance the subject selection elements of your style.
I never really got caught up too much in worrying about style. I'd read about it, listen to what other photographers had to say about it. I always cared about whether I liked the way my images looked rather than what others thought of them.
Photographers who practice photography over the course of years will come to develop their own styles whether they're conscious of it or not, whether intentional or otherwise unplanned. It's not something to worry about - it's inevitable, like death.
As a photographer, it's important however to think about the elements of style, the concrete details which the photographer are able to shape at present. These are the essential details which define you at any given point in time. It's the short term journey. Style is the long haul.
And if it's true what some photographers say, that you're only as good as your last image, then the photographer should be focusing on the elements of style and relegate all talk of style to unheated conversations and one-liners in bios.