There are a million definitions of a brand.  Merriam Webster defines a brand as "a public image, reputation, or identity conceived of as something to be marketed or promoted." 

The American Marketing Association weighed in with “A name, term, design, symbol, or any other feature that identifies one seller’s good or service as distinct from those of other sellers."

While Jeff Bezos has been famously quoted as saying "our brand is what other people say about you when you're not in the room."

I believe a brand is all of those things and then some. A brand, in my estimation, is 

the impression that you leave about who you are and what you stand for. 

That impression is formed over time through all the contact points a person has with your brand. It's influenced by their customer experience, your website navigability, the tone and placement of your messaging, and, of course, through the design esthetic, use, and consistency of your visual collateral (primary example: your logo). 

Identity design is what many people think of when they think of branding. It is the intentional creation of brand assets, including logos, favicon, brand marks, presentation and social media templates, and much more. It is essentially 

the visual representation of your brand.