Hello OP! The Jacobites wanted the dethroned James II and/or his descendants (the House of Stuart) restored to the thrones of England, Ireland and Scotland. "Jacobite" comes from the Latin version of James' name, "jacobus."

James II was Catholic, whereas the majority of English elites were protestants. James' two eldest daughters from his first marriage to Anne Hyde, Mary and Anne, were also Protestant.

However, James' second wife was the Catholic Mary of Modena and, in 1688, they gave birth to a son, James Stuart. James would be raised Catholic and would come before his Protestant sisters in the succession.

This, especially when combined with James' absolutist tendencies, was intolerable to many English elites. After securing English Protestant support, James' eldest daughter Mary and her husband William of Orange invaded England, forcing James II and his family to flee to France. The relative ease of this transition, especially when compared to the English Civil War, led this event to being called "the Glorious Revolution."

The problem is that there were still Catholics, especially in Scotland, who believed James and his line were the rightful rulers and who were persecuted under English Protestant rule. Furthermore, James II and his descendants had the support of other Catholic European nations, such as France, Spain and the Papal States. This led to several rebellions and invasions, some thwarted early and other very violent, through the first half of the 18th century. The most famous of these is the Jacobite rising of 1745, which involved James' grandson Charles Stuart ("Bonnie Prince Charlie") raising an army in Scotland before being defeated at the Battle of Culloden.

After this, Jacobitism declined for several reasons, including greater English oversight and control of the Scottish clans, the loss of French support, and, finally, the death of the last direct male Stuart heir - Henry Stuart, styled Henry IX - in 1807.

I did want to add a note about the impact of Jacobitism. The Jacobite risings were incredibly important in the history and cultural memory of Scotland, and act as a backdrop to lots of novels and songs. Sir Walter Scott set several works during or around the Jacobite risings, including Waverley and Rob Roy. There are also dozens of Scottish songs inspired by or referencing Jacobitism, such as "The Skye Boat Song" and "Hey, Johnny Cope, Are Ye Waking Yet?" This literary and folk history has kept Jacobitism in the public consciousness, and might explain why there are so many references to it without further explanation.

Finally, as I am not a historian, so I should point out an excellent answer by /u/lngwstksgk here that gives more details about the Glorious Revolution and its aftermath.

Sources

Diana Preston. The Road to Culloden Moor: Bonnie Prince Charlie and the 1745 Rebellion. London, UK: Endeavor Press, 1994. Szechi, Daniel. 1715: the Great Jacobite Rebellion. Yale University Press, 2006. Peter Ackroyd. Rebellion: The History of England from James I to the Glorious Revolution. New York: Thomas Dunn Books, 2014.

Perhaps a quick note on the difference between "Jacobite" and "Jacobin" might be in order. I know I was confused when I first started to study history.

The Jacobites were supporters of the deposed King James II of England and VII of Scotland, of the Stuart house.. He was deposed in the 17th Century from the British throne in favour of William of Orange. This was at the time of a lot of Catholic / Protestant bloodshed, and James was Catholic where William was Protestant. Similarly England was largely Protestant, while Scotland was largely Catholic. As such Scotland had a large amount of support for the deposed James, whereas England largely supported William. The name Jacobite comes from Jacob, the latin for James.

With James fled to the continent, there were a number of Jacobite movements over the 18th century which aimed to replace the Stuarts on the throne. Although this was mainly a Catholic / Protestant issue, with the aforementioned regional split of religious support it was also largely a Scottish / English point.

A number of events that are quite famous in British history occurred around this time, linked to the Jacobites.

Since James had strong support in Scotland, when he took the throne William ordered that the Scottish clans pledge allegiance to the throne. The Macdonald clan in Glencoe was (allegedly) late in pledging allegiance. Troops from the rival clan Campbell were ordered to make an example of them. The Campbell troops arrived as friends, were quartered by the Macdonalds, and then during the night rose up and slaughtered their hosts. Rather than scaring the clans into line, ultimately this act of barbarism increased Scottish support for James.

There were then a few rebellions – the main ones I can think of were in 1715 and 1745. The rebellion in 1745 is perhaps the most well-known.

The 1745 rebellion was led by "Bonnie" Prince Charlie (grandson of James II and VII). He landed in Scotland to summon an army from the largely Catholic Scots who were sympathetic to the Jacobite cause. The British armies were largely fighting the French at the time and had to be recalled to defend. The Jacobite army made it as far south as Derby, however they weren't getting as many reinforcements from the English Catholics as they had expected / hoped and were concerned about news that the Duke of Cumberland had landed back in England with his army so they turned back north on around 6 December 1745.

The Duke of Cumberland effectively chased the Jacobite army the length of Britain until they met at the Battle of Cullodden on 16 April 1746. The British government army tactics and equipment ultimately won the day. The Jacobite battle plan was based on the Scottish clan method of a charge with sword and shield, however the ground wasn’t conducive to rapid charges being uphill for them and lumpy. The British government army however stood firm with muskets and grapeshot cannons. The battle was a decisive victory for British forces under Cumberland, and Bonnie Prince Charlie fled Scotland. There is still lots of high feeling in Scotland over the battle - for example claims that Cumberland ordered his soldiers to put down the Scottish wounded, to the extent Cumberland has the nickname "The Butcher" in Scotland.

Cullodden has the additional claim to fame of being the last pitched battle fought on British soil.

As could be expected following this uprising (and the earlier ones) there was a backlash from the British government against the rebels. Given the strong Scottish support for the Jacobites, and in particular from the clan based highlands, this backlash took a strong anti-Scottish slant.

Following this, when Scottish clansmen would raise glasses to toast "the King", they would pass their glass over a bowl of water on the table, signifying the King over the water - ie King Charles (Bonnie Prince Charlie) who was on the Continent. As a side note, as a result for a long time the British royalty did note have finger bowls on the table, to avoid anyone doing similar in their presence.

Source: Educated in Scotland, which focused heavily on that period

The Jacobites were those people in Ireland, Scotland and England who refused to accept the 'Glorious Revolution' of 1688 and the rebalancing of power between the Aristocracy, Gentry and Monarch of the Kingdoms of Ireland, England and Scotland. They were loyal to the house of Stewart rather than to the house of Hanover.

The Glorious Revolution was the invasion of England by William of Orange who was hereditary 'Stadtholder' or Steward of most of the modern day Netherlands and was married to Mary Stewart (Mary II of England). William and Mary ruled together and their reign is usually called William and Mary.

William of Orange was invited to invade England by a section of the English aristocracy and gentry who were concerned that the Catholic King of England James II was becoming increasingly autocratic. Some powerful English people were also concerned that if their king was a Catholic they would become subservient to France. They could tolerate a Catholic King but not a Catholic dynasty so when James II had a son they invited William who was married to the former (protestant) heir presumptive Mary to invade.

It is widely believed that most people in England as well as in Scotland and Ireland initially supported James II. However William was soon popular in England.

In 1715 Queen Mary's sister Anne (Queen Anne died) and the elector of Hanover was made George I of England.

In general there are three reasons for Jacobitism.

Religion -- James 2nd and 7th was religiously tolerant towards Roman Catholics. This is one of the reasons that he was overthrown. William was religiously tolerant but was not allowed to extend this tolerance towards Roman Catholics. Most of the Irish gentry pre 1690 was Catholic and also rather surprisingly so was most of the lowland Scottish gentry. In general one could participate in British controlled Ireland by becoming Protestant and many did so but not everyone wanted to.

Opportunism For the Irish gentry and aristocracy especially opportunities were restricted under the Williamite / Hanoverian order. There were quite restrictive laws regarding what one could do, property that one could hold, horses etc. Certainly in terms of a military career continental Europe offered more opportunity and here one would have to be a Jacobite/ Catholic.

Highand Inter Clan Politics In general we can call the story of the Scottish highlands - the Campbells and the rest (the Campbells win) the Campbells quickly accepted the new order especially after William guaranteed rights for Calvinists. In this sense Jacobitism is a resistance to the Campbells as much as to the English.