The Finches do not live in Montgomery, the capital of Alabama; instead they live about 100 miles south of Montgomery in Maycomb (county seat of Maycomb County) which is the fictional name for Harper Lee's hometown of Monroeville, the county seat of Monore County. This city is located near the Alabama River where there was once travel downstate. After the railroad came to this area travel changed. This railroad is the one that Dill travels to and from Meridian, Mississippi.
In Chapter 13, after Aunt Alexandra arrives, Scout learns that the Finches originally lived by the river on Finch's Landing. But, after a certain man named Sinkfield, who had the town's boundaries changed to be more inland and encompass his tavern, they moved into Maycomb, which had sprung from Sinkfield's Tavern. As the largest city in the area, Maycomb became the county seat. However, during the Civil War, there were no battles fought in the area, probably because the population of the area was small. This Civil War is what is referred to as the "disturbance between the North and the South."
As indicated in the previous post, the Finches did not come to Montgomery. Atticus went there to study law. Mrs Finch, however, was born and bred in Montgomery and Atticus met her when he was first elected to the state legislature. He was already in his middle ages by then and she was fifteen years his junior. Jem was born a year into the marriage and Scout four years after. She died of a sudden heart attack two years later. These are the only real associations the Finches have with Montgomery and they never went to live there.
On his admission to the Bar, Atticus returned to Maycomb where he opened up a law practice. He established his family there and they lived there ever since.
The reference to 'the disturbance between the North and South' alludes to the American Civil War which was a conflict between the Northern and the Southern states primarily brought about due to their differences on the issue of slavery and the matter of extending it to the western regions. Seven Southern states seceded from the Union and formed the Confederate States of America.
The war began in April 1861 when Fort Sumter, a US fortress, was attacked by the Confederates. The war lasted four years and is deemed the most bloody in American history. It ended with the collapse of the Confederate government and the surrender of all the Confederate armies in the spring of 1865. In the process, much of the South's infrastructure had been destroyed, which would explain Simon Finch's anger, as mentioned in the following extract:
Simon would have regarded with impotent fury the disturbance between the North and the South, as it left his descendants stripped of everything but their land ...