In the Scottish church of the 18th and 19th centuries, a burgher was a person who upheld the lawfulness of the burgess oath.

The burgess oath was the oath which a  town burgess was required to swear on taking office. 

The burghers' position was in opposition to the  seceders and Anti-Burghers.

Background
The Rescissory Act 1661 had required stated that all ministers and preachers in Scotland needed to acquire a patron (usually a local laird who would choose which minister would preach in their area).  A quarter of the clergy refused to hand over this authority to a person outside the church and consequently lost their jobs. They continued to preach independently and illegally, which led to armed rebellion and The Killing Time in the 1680s. Patronage in Scotland was halted in the 1690s.  

A new  Patronage Act was legislated in 1711. According  to Dale Jorgenson, "...The Patronage Act, enacted under the reign of Queen Anne (1702-14), gave lay patrons the right to present ministers to parishes. This act of patronage was an affront to classic Presbyterianism, and resulted in a division between Burghers who accepted the Burghers' Oath and its consequent patronage, and the Anti-Burghers who would not accept the oath."

Church splits
The First Secession occurred in 1731 and was led by the General Assembly of the Church of Scotland. It stated that church ministers should be chosen by church elders. This led to the creation of the Associate Synod in 1742. 

The secession church split in 1747 into the Burghers and the Anti-Burghers over the lawfulness of the forms of the oath then current in Scotland.  The contentious clause required the burgess to profess that the true religion was the one professed within the realm. 

The Burghers stayed with the Associate Synod, while the Anti-Burghers created the General Associate Synod. 

Both groups later had internal splits, with the Burghers splitting in 1798 into the ‘Auld Licht’ Calvinist group, which held to the  Solemn League and Covenant, and the ‘New Licht’, which was more liberal and influential.  The Auld Lichts created the Original Associate Synod.

Legacy of Burghers

In 1820, many of the Burgher and Anti-Burgher congregations united into one denominations. Some churches did not wish to unite and went on to form a separate church denomination.

In 1842, the General Assembly of the Church of Scotland wrote to the newly-crowned Queen Victoria, urging the end of patronage. This did not happen and several ministers went on to form the  Free Kirk.   

Patronage was finally abolished by Parliament in 1874, after 300 years.

Notable Burghers
Robert Balmer ('New Licht' - subsequently Professor of Theology of the United Secession Church)
John Brown of Haddington (ordained after the 1747 'Breach' and died before the 'Auld Licht'/'New Licht' division), Professor of Theology of the Associate Presbytery
John Dick, 'New Licht' Professor of Theology of the Associate Presbytery 
Ebenezer Erskine (one of the 1733 seceders) (died before the 'Auld Licht'/'New Licht' division)
Ralph Erskine (seceded 1737) (died before the 'Auld Licht'/'New Licht' division)
James Fisher (one of the 1733 seceders) (died before the 'Auld Licht'/'New Licht' division)
George Lawson (1749-1820), 'New Licht' Professor of Theology of the Associate Presbytery. 
Michael Willis 'Auld Licht' Professor of Theology, son of the next
William Willis, 'Auld Licht' Professor of Theology, father of the previous

Theological Professors

Before the 'Auld Licht'/'New Licht' division (1747-1800) 

1. James Fisher (1749-1764)

2. John Swanston (1764-1767)

3. John Brown of Haddington (1768-1787)

4. George Lawson (1787-1800).

New Light (1800-1820) 

1. George Lawson (1787-1800).

2. John Dick (1820)

Old Light (1800-1839) 

1. William Willis (1800-1803)

2. George Hill (1803-1819)

3. William Taylor (appointed interim Professor, 1818) (1819-1833) (died 1836)

4. Michael Willis (1835-1839)

References
Citations

Sources
 

Presbyterianism in Scotland
Schisms in Christianity
History of the Church of Scotland
18th century in Scotland
19th century in Scotland
18th-century Calvinism
19th-century Calvinism
Church of Scotland