A brand is a collection of characteristics (not necessarily visual) that signify an entity. You can brand an individual; a city; an event, like the Olympics or a wedding; a business; or a campaign.
Your brand should be accurate and in line with expectations. For example, bright, cheerful colors belong with a preschool, not a funeral parlor.
Branding is a two-way street. If you don't live up to your brand, you could taint it. A smiling cartoon won't help a business with terrible customer service. And irony can turn a brand into a mockery.
In short, your brand is your avatar—make each other proud!