As I understand the term, an off license is a business where alcoholic drink can be purchased to be consumed 'off' the premises. People buy drink there to take home or to somewhere else (a carry out). The license refers to the fact that alcohol can be bought/sold there but cannot be comsumed 'on' the premises.
You apply for a license from Birmingham City Council using forms you can download from the Department of Culture, Media and Sport. http://www.birmingham.gov.uk/GenerateContent?C ONTENT_ITEM_ID=41596&CONTENT_ITEM_TYPE=0&MENU_ ID=13603
Cost? - depends on the non-domestic rateable value of the premises licensed to sell the alcohol, but between �100 and �600 for the initial application then an annual fee of about two-thirds of these figures.