An official is, in the primary sense, someone who holds an office in an organisation, of any kind. For example, in baseball, the official scorer is a person appointed with a duty to keep the score, and make a definitive record. The term officer is close to being an synonym (but has more military connotations). A functionary is someone who carries out a particular role within an organisation; this again is quite a close synonym for official, as a noun, but with connotations closer to bureaucrat and so sometimes mildly pejorative in English usage. Any such person acts in their official capacity, in carrying out the duties of their office; they are also said to officiate, for example in a ceremony. A public official is an official of central or local government.
As an adjective, official often but not always means pertaining to the government, either as state employee or having state recognition. For example, an official holiday is a public holiday, having national (or regional) recognition. An official language is a language recognised by a government, for its own use in administration, or for the use of citizens (for example on signposts). An official spokesperson would be an individual empowered to speak for the government, or some part of it such as a ministry, on a range of issues and on the record for the media. An official statement is issued by an organisation as an expression of its corporate position or opinion; an official apology is an apology similarly issued by an organisation (as opposed to an apology by an individual). Official policy is policy publicly acknowleged and defended by an organisation. In these cases unofficial is an antonym, and variously may mean informal, unrecognised, personal or unacknowleged. An official strike is a strike organised and recognised by a labour union, as opposed to an unofficial strike at grassroots level.
An official history, for example of an institution or business, or particularly of a war or military unit, is a history written as a commission, with the assumption of co-operation with access to records and archives; but without necessarily full editorial independence. An official biography is usually on the same lines, written with access to private papers and the support of the family of the subject.