Dukes are formally referred to as 'The Duke of (place name)', and addressed as "Your Grace". On State occasions, dukes rank below Royal Dukes, and above Earls. This is called the order of precedence, because in procession to the opening Parliament, or on other state occasions, they walk in order of their rank.
The office or position of a duke is called a dukedom.
The wife of a duke is a Duchess. This is an honorary title, given for life to the lawfully wedded wife of a Duke. If she divorces and remarries a commoner, (person with no title) their children have no title. The children of a duke, on the other hand, do have honorary titles (so long as they are born in lawful wedlock). The eldest son may bear what is called a junior title which an ancestor was given before the dukedom was created. Other children would be called 'Lady' Jane (or other Christian name) and 'Lord' James (or other Christian name). Children use the family surname, not the Dukedom.