So let's start at the top.

Sir Arthur Wellesley, 1st Duke of Wellington on the left.
Sir Arthur Wellesley began his military career in the British Army in 1787, serving in Ireland, before being elected to the Irish House of Commons. He gained his colonelcy in 1796 and took part in campaigns in the Netherlands and India. His success in India gained him a governorship there. In 1805 he returned to the UK being quoted as saying "I have served as long in India as any man ought who can serve anywhere else." During his journey home he stopped at the Isle of Saint Helena and stayed in the same building that would later be used as a prison for Napoleon Bonaparte.
After arriving in the UK he took part in battles in Northern Germany and Denmark. He was due to sail with 9000 troops to South America but was diverted to the Iberian Peninsula. (Portugal and Spain). He won his first two engagements against the French in Portugal, but when his superior officer General Dalrymple arrived, he not only released all of Wellesley's prisoners, but agreed to ship them and all their loot back to France. Recalled to England, Wellesley was cleared of misconduct charges, though General Dalrymple was not. Between 1808 and 1813 the British army along with their Portugese and Spanish allies campaigned back and forth across the peninsula. Wellesley became very good at choosing his fights and picking out the ideal spot to deploy his men. Whilst the allied forces didn't win every engagement, Wellesley eventually defeated every French Marshal sent against him.
After Napoleon abdicated in 1814 Wellesley was given the title the Duke of Wellington, the first to bear such a title and the name he is most commonly referred to by. The duke was assigned as Ambassador to France where he helped sort out the details of France's disarmament, arguing strongly that France should be allowed to maintain it's position in the European balance of power.
In 1815 Napoleon returned to France and immediately began to rebuild his army. England, Russia, Prussia and Austria each agreed to send 200,000 men to stop him. Wellington was given overall command of the Allied British, Dutch and German contingent and took to the field alongside General Blucher's Prussians. Napoleon's army was eventually defeated after the battles at Quatre Bras, Ligny, Wavre and Waterloo. Napoleon threw himself on the mercy of the British and was imprisoned at Saint Helena where he later died.
After the war Wellington became an increasingly political figure. In 1828 he became Prime Minister of England for the Tory Party. During his time in power he was influential in establishing King's College in London and helped push through the Catholic Emancipation Act. (Prior to which catholics were prosecuted by the largely Protestant British government at the time). It was during this time that he gained the nickname "The Iron Duke," largely due to his staunch opposition of several attempts to reform the government. In 1830 his government was disbanded, but later regained power in 1834. During the 1834 election, Sir Robert Peel was elected Prime Minister but was away in Italy at the time. For 3 weeks Wellington was the Acting Prime Minister.
Upon his death in 1852 Wellington was granted with a state funeral, something that only a handful of British subjects have been honoured with. (Lord Nelson and Winston Churchill being two of them). He was interred in St. Paul's Cathedral next to Lord Nelson, whom he had only met once, briefly just before Nelson was killed at the Battle of Trafalgar.
Next.... Sir Thomas Picton