Ok history lesson. Buckle up - it is complicated.
At the Reformation, the church in Scotland became Presbyterian the established Church of Scotland.
During the 17th Century the Stuart monarchs tried to impose an Episcopalian system and prayer book on the Scottish church. There was a lot of opposition to this and battles fought. Many people who didn't want to submit to the imposed system met in the open air with ministers who had been expelled from their churches. There was a lot of politics involved as well as faith, but people did suffer and were martyred for their faith.
The situation was eventually resolved by the settlement under King William and Mary which reestablished the Presbyterian church in Scotland. Some of those who had been most opposed and suffered most during the 'Killing Times' remained out of the Established church and they became the Reformed Presbyterian Church of Scotland.
During the 18th century various other groups left the Established Church of Scotland. The issues were varied but mainly concerned state interference in church government and feelings that the Established Church was not maintaining true doctrine. The most enduring of these groups were the Associate Presbytery and the Relief Church. There were further divisions and reunions in these groups but eventually in the 19th century they had mainly formed the United Presbyterian Church. This bit is simplified a lot!
The problems about state interference and differences in theology continued in the Established Church and this led to the Disruption of 1843 which cottagecheese has explained above.
Initially the Free Church had no churches or manses and, as the schools were run by the Established Church, Free Church school teachers also lost their jobs and homes.
Gradually the Free Church built new churches, manses and schools (and teacher training colleges - most of the Education departments in Scottish universities today were originally Free Church foundations).
In 1876 most of the Reformed Presbyterian Church joined the Free Church but some remained separate. They are now very small and strongest in southwest Scotland which was their original heartland.
Parts of the Free Church became more liberal in theology towards the end of the 19th century and there was also a movement to unite with the United Presbyterian Church. There were also groups opposed to both movements. In 1893, when the union seemed imminent, a group left the Free Church and became the Free Presbyterian Church of Scotland.
The union of the Free Church and the United Presbyterian Church eventually did take place in 1900 and became the United Free Church.
However, a minority remained as the Free Church of Scotland. Some of the Free Presbyterians rejoined the Free Church then, but others remained as the Free Presbyterian Church. There isn't really any doctrinal difference between the Free Church and the Free Presbyterians but the FPs have remained much stricter in practice and the recent changes in Free Church practice have widened this gap.
In 1929 most of the United Free Church rejoined the Church of Scotland as parliamentary legislation had resolved the state interference issues. Again a minority stayed out and still exist. I have to admit that I don't know much about them.
In 1989 the Free Presbyterian Church divided over the issue about Lord Mackay as mentioned previously and the Associated Presbyterian Churches were formed. They are a little less strict in practice.
In 2000 the Free Church divided and the Free Church Continuing was formed. There were disagreements about how a case involving an individual minister was handled but there was also an underlying unease about some movements to change practice within the Free Church. The minister concerned was made something of a focal point.
This is recent history and divided friends and families and is still a cause of pain.
The Free Church Continuing were however correct that changes were afoot in the Free Church and these (mainly concerning the form of worship) have now come to pass.
Congratulations if you got to the end of that!!