Simply put, branding is the practice of communicating your professional image through visual design. Let's use PRIMARK as an example: Their brand identity feels affordable, and that's entirely on purpose. They don't want to be a high-end fashion brand: They want to be the first clothing retailer on someone's mind if they need cheap clothing.
Meanwhile, designer brand stores can only be visited on an appointment basis: They want people to know that the average bypasser could never afford to shop there. It's all in the brand, and it's all by design.
Most of us rely heavily on our eyes to get around. Visuals give us all sorts of clues. What do you look for if you're on holiday and need a pharmacy? Probably one of those seizure-inducing green 'plus' signs. This principle can be applied to all sorts of businesses: The right look lets consumers know what you're about in a snap second– and that's important because a snap second is often all you get.
You may not need a visual identity if you're the only source in town for the thing(s) you offer. Unfortunately, that's unlikely. With many businesses fighting for attention, you must maximize your potential by being recognizable. Try to think of a strong company that doesn't have a highly recognizable logo or color scheme. Pretty tricky, right?
A low price sensitivity means that your target market's willing to pay what you're asking for your product or service. It means they like your brand well enough to give less weight to price. A strong brand enables you to raise prices because you've established your business as the go-to in your industry in the minds of your target market.