Next to just a name, term, design and symbol I also feel that branding is an experience or a feeling that a customer gets from your brand.
A brand can be anything, a product, a company or even yourself.
To me, Branding is:
The experience or the feeling you try to give to people about your brand.
People will have formed an opinion about your product, company (etc.,etc.) that, if done correctly, corresponds with the message you try to put out.
Successful branding means that people will understand your brand and (preferably) choose it as their favourite. When branding is successful they will pick your brand from your competitors because the feeling you put out about your brand corresponds with theirs.