A pompous person is arrogant or conceited. He'll walk into a party with an inflated ego, ready to tell anyone who will listen that "I'm kind of a big deal."
Today we associate the adjective pompous with self-important jerks. But it's actually derived from the Old French pompeux, which meant “stately." And that's why you can also use pompous to describe something with a lot of ceremonial or stately display — in other words, something surrounded by "pomp and circumstance."