Short answers, pulled from the story.
The Duke of Wellington title was created on the 3rd of May 1814 for Arthur Wellesley, the Anglo-Irish military commander born in 1769. It was conferred by the Prince Regent on behalf of George III after Wellesley returned home following Napoleon's abdication.
The title derives from Wellington in Somerset. Arthur Wellesley's brother Richard identified a manor in the parish of Wellington as available for purchase, noting it was reasonably close to the family surname, and arranged the acquisition while Arthur was still commanding the army in Spain.
The Dukes of Wellington hold the title of Prince of Waterloo in the Netherlands, Duke of Ciudad Rodrigo with Grandeeship in Spain, and Duke of Victoria in Portugal, along with the subsidiary Portuguese titles of Marquess of Torres Vedras and Count of Vimeiro. These were granted to the 1st Duke for his victories in the Peninsular War and at Waterloo.
The 9th and current Duke of Wellington is Arthur Charles Valerian Wellesley, born in 1945, who succeeded to the title in 2014. He is married to Princess Antonia of Prussia.
If the direct male line from the 1st Duke becomes extinct, the British dukedom and the Dutch princeship of Waterloo would become extinct. The Spanish dukedom of Ciudad Rodrigo would continue in the female line, while the Irish titles of Mornington and Wellesley would pass to the line of the Earl Cowley.
The family seat is Stratfield Saye House, near Basingstoke in Hampshire. Apsley House in London is now owned by English Heritage, though the Wellington family retain an apartment there.