When analyzing sociological background, one is analyzing the type of society that the author is writing about. You are analyzing things like the types of government, types of laws, and types of rules. You can also analyze if you think the author actually likes the society, or does he/she dislike it. This can also lead you to analyze what social changes the author might make.
Jane Austen's Pride and Prejudice is a very true sociological reflection of Austen's own social situation during the Regency period. Just as Austen knew it, England's government in Pride and Prejudice is a monarchy that is governed upon the advise of the Prime Minister, Cabinet, and members of Parliament. Also, just as Austen new it, her novel's society is divided by class rank, including the noble class, the landed gentry, the middle class, and the merchant class. In fact, one social phenomenon Austen experienced herself and wrote about in the book was the increase in wealth of the merchant class. Around the same time as the French Revolution, some members of the merchant class were becoming increasingly wealth, wealthy enough to rub shoulders with the landed gentry, purchase estates of their own, and live among the gentry. At the same time, the English nobility watched all of the French nobility be beheaded. Not wanting the same fate, the English nobility decided to begin softening some of the lines of demarcation separating classes. Austen's era began seeing the intermarrying of the classes ("Social Evolution"). Both the rise of the merchant class and the mingling of the classes are social issues Austen covers in her novel.
Austen especially portrays the social issue of the rising merchant class with Mr. Bingley. Austen makes it evident that Bingley's father actually earned the Bingley fortune through trade and left it to Bingley to buy the family estate. However, using Mr. Bingley and his sisters, Austen points out the consequence from the social phenomenon, which was in increase in prejudicial snobbery. In particular, we see Bingley's sisters criticize the Bennet family, which are actually in a class higher than they are since Mr. Bennet is gentleman with an estate, simply because Mrs. Bennet is descended of the working class, just like the Bingleys. As Austen phrases it:
[The Bingley sisters] were ... in every respect entitled to think well of themselves, and meanly of others. They were of a respectable family north of England; a circumstance more deeply impressed on their memories than that their brother's fortune and their own had been acquired by trade. (Ch. 4)
Hence, one aspect of society Austen shows a desire to change is class snobbery.
In addition, Austen also portrays inter-class marriages when she has Elizabeth marry Mr. Darcy. Technically, they are members of the same class. As Elizabeth points out to Lady Catherine de Bourgh, "In marrying your nephew, I should not consider myself as quitting that sphere. He is a gentleman; I am a gentleman's daughter; so far we are equal" (Ch. 56). However, some, like Lady Catherine, may consider Darcy to be a cut above Elizabeth because he has noble class relations while Elizabeth, again, has working class relations. Austen is not only using the marriage of Elizabeth and Darcy to support inter-class marriages, which were on the rise in her society, she is also using Lady Catherine to once again criticize class snobbery.