Jimjools is correct - British citizens need a passport as it is the only official id card issued by the UK government. When we get id cards they will be sufficient.
Many European countries are signatories to the Schengen agreement, which means that there are no border controls between them. For example, you don't need any documentation, not even an ID card, to travel between France and Belgium.
However, the UK refuse to sign up to the agreement, which means that you need a passport to visit any other country, with the sole exception of the Republic of Ireland.