I'll try and answer this one simply,though there is much depth to understanding the originality of brands. The following isa sketchpad overview, and extension of the post: The 3 Elements of a Great Brand
Brands are unique when they inspire, compell to action, and communicate valuable differentiation.
When they are accurately positioned in the marketplace via a fresh voice and inspired message, focused around a trend.
So, let's say you sell organic dogfood. Some elements of your brand might align with: health for your pet, good taste, perceived affluence, and earth friendly manufacturing practices. If no other brand is championing these things, and you are certain that your target market will find them appealing, then you have the start of a unique brand.
You then develop a mantra for your co./brand that epitomizes those critical elements of difference, then translate that into a unique message that conveys your point of difference in a remarkable way (as simple as a tag line, or broader communications that tell your story). While only a small part of "branding", this is the unique message you want your target market to remember, thus quite important.
The brand and company need a clear spirit and personality to display to the world when showcasing those brand elements; something that matches what your audience will be receptive to while providing flexibility for communicating each of the chosen brand elements.
Identity is critical, so a name and logo that match your brand promise and personality must be crafted with care (more about this another day).
Now, here's the scary part.
This is all simply a smart "strategy" to approach the market with.
Ultimately, the masses decide if your brand is unique, and it's your job to steer them to understanding via an unwavering focus on delivering brand experiences that resonate with your core message, personality and promise.
If you do this in creative, unpredictable (but consistent) ways, you'll have a better shot than most at developing a unique brand.