Lady Catherine de Bourgh is a difficult character to fully understand. Her role in Pride and Prejudice is complex. Yes, she is arrogant, pompous, intrusive and annoying, yet she is also a woman with extensive authority and power in an era when women were as a rule without authority and power. Married women in Austen's era were legally an extension of their husbands; they were in a sense no more important legally than a household domestic, although, when you consider it, even single female domestic workers could enter into contracts, such as their employment contracts (a point illustrated so well by Hardy in Tess of the d'Ubervilles), whereas any married woman could not.
This was according to the English legal principle of couverture which distinguished between feme solo (unmarried) and feme covert (married). The difference between the two designations was that single, feme solo, women could legally own property and enter into contracts while married, feme covert, women could perform neither of those legal activities. The reason is that a woman's identity was legally subsumed under her husband's identity upon marriage.
Then here you have Lady Catherine. Austen's narrator makes a point of informing us that Sir Lewis de Bourgh's family saw no need to neglect the female line of the family and render it powerless: Sir Lewis's will had specified that Lady Catherine de Bourgh was to inherit all his estate and all the power and authority that went with it.
[Lady Catherine says,] "I see no occasion for entailing estates from the female line. It was not thought necessary in Sir Lewis de Bourgh's family."
And, yes, she greatly enjoyed the exercise of that power and authority. Remember, though, that, regardless of her abrasive personality, she attempted (successfully or not) to use that power for good amongst her villagers to help them be prosperous and lead peaceful lives.
Elizabeth soon perceived ... [that] this great lady ... was a most active magistrate in her own parish, ... and whenever any of the cottagers were disposed to be quarrelsome, discontented, or too poor, she sallied forth into the village to settle their differences, silence their complaints, and scold them into harmony and plenty.
Austen uses Lady Catherine to make a case for female legal identity and power. Yet by showing that Lady Catherine's unusual power is still exercised in accordance with the cultural prejudices prevalent in their society, as seen during her confrontation with Elizabeth about a rumored engagement to Darcy, Austen also uses Lady Catherine to illustrate the theme of equality of personhood in love and marriage (see Themes Insights: Equality of Personhood).
While Lady Catherine's characterization challenges the legal prejudices against women, her character also shows uncompromising cultural prejudice against people in lower classes, namely Elizabeth's mother, aunts and uncles who are of a lower class:
"True. You are a gentleman's daughter. But who was your mother? Who are your uncles and aunts?"
Lady Catherine is a complex character who illustrates (1) Austen's social protest about the legal position of women, (2) Austen's social protest about prejudice against lower classes, and (3) Austen's interesting theme relating to what affects the equality of personhood in family, love and marriage.