An accent is simply how one pronounces words—a style of pronunciation. A dialect includes not just pronunciations, but also one's general vocabulary and grammar.

Can you have a dialect without an accent?

Speaking with "no accent" is the same like not speaking in a dialect: it means you speak the dialect which has become the "official" one. No, everyone has an accent. They just think that they have a "neutral" accent, because that's the way they expect everybody should talk.

Is British an accent or a dialect?

Most people in Britain speak with a regional accent or dialect. However, about 2% of Britons speak with an accent called Received Pronunciation (also called "the Queen's English", "Oxford English" and "BBC English"), that is essentially region-less.