The political background of Outlander has to do with the conflict over Jacobitism. The origins of the conflict go back to 1603 when Queen Elizabeth I of England died without any children to inherit her throne. The English crown then went to her next closest relative, King James VI of Scotland (James was the great grandson of Elizabeth's aunt, Margaret Tudor, the sister of King Henry VIII of England). James' coronation inaugurated the reign of his house, House Stuart, over England. While England and Scotland shared the same monarch, they maintained separate Parliaments and were technically separate countries.
The reign of the Stuarts was a tumultuous one and James' son and heir, King Charles I, was deposed and executed during the English Civil War in 1649. From 1649 until 1660, England was ruled by Parliament under the 'protection' of Oliver Cromwell in what has become known as the 'Interregnum', but with the death of Cromwell, James grandson, Charles II was restored the monarchy. When Charles II died in 1685 leaving no legitimate heirs, his brother, James II succeeded him. King James' rule quickly became controversial.
In the 17th century conflicts between Protestants and Catholics continued to roil Britain. Many Protestants feared that the Catholics were trying to take over England and Britain. James II had converted to Catholicism and married an Italian princess, Mary of Modena. At the time of his coronation this was tolerable to most English Protestants because James' only heirs were his Protestant daughters by his first wife, Anne Hyde. However, after James' coronation, his wife gave birth to a son (James III/VIII). This son would be raised Catholic and would take precedence over his elder sisters in the line of succession. Suddenly, English Protestants were faced with the very real possibility of a lasting Catholic monarchy in Britain. They were also angered by James's efforts to extend greater religious freedom to British Catholics.
In December 1688, concerned members of Parliament sent a message to James' eldest daughter Mary, asking her and her husband, the Dutch prince William of Orange, to come to Britain and assume the throne in order to protect the Protestant faith. In an episode which became known as "The Glorious Revolution", William landed in England with the Dutch army and deposed James II in a nearly bloodless revolution. William and Mary became co-monarchs of England and Scotland.
While James fled to Europe (throwing the Great Seal of Britain into the Thames along the way), not all Britons were reconciled to the end of the Stuart monarchy. Those who still supported the Stuarts' claim to the throne became known as Jacobites, since they supported the claim of James II and his descendants. Over the half century or so between the Glorious Revolution and the events of Outlander, there would be periodic attempts by the Stuarts to raise rebellions in their favor - most notably in 1715 and 1719.
The Highlands of Scotland, where Outlander is set, was a particular hotbed of Jacobite sympathy, with many Highland lords feeling that the Stuarts had treated them better than their Hanoverian successors had.
It's important to note that while the first season of Outlander often seems to portray the conflict in the Highlands as a nationalist one between Scots and English, the reality of the Jacobite and anti-Jacobite conflict was not national in nature. There were plenty of Scots at the time who opposed the Jacobites and supported the continued rule of the Hanoverian dynasty of King George II and there were notable English Lords who supported the Jacobites as well.