This page should offer some basic insight on the matter. You apply to become a British citizen through a general form; having had background checks performed on your health, education, financial status and so on you will either be permitted or disallowed to live in Britain and then it is after having lived there for five years (or two years if married to a British citizen) that you shall achieve full citizenship, that is, you will be able to vote in official British elections (EU elections i’m uncertain of), be called up for jury duty, receive certain financial benefits, acquire a shiny red passport etc.