Literary style is the manner in which one author writes that is unique from other authors. What can help characterize individual style are things like, sentence structure, grammatical choices, vocabulary, tone, point of view, and figurative language, as well as many other details ("Checklist: Elements of Literary Style"). What your assignment is asking you to do is take any passage and analyze it to characterize Austen's style. Since we are limited to space, we won't be able to discuss all the details of style listed above, but here are a few ideas to get you started.
Let's take a look at the opening lines of the novel since they actually are quite unique and revelatory:
It is a truth universally acknowledged, that a single man in possession of a good fortune must be in want in want of a wife. However little known the feelings or views of such a man may be on his first entering the neighbourhood, this truth is so well fixed in the minds of the surrounding families, that he is considered the rightful property of some one or other of their daughters. (Vol. 1, Ch. 1)
One thing you might notice right away is that the second sentence is certainly exceptionally long. It is also a complex sentence in that it has what we would grammatically call a complex sentence structure. It begins with an introductory clause, followed by an independent clause, followed by yet another dependent clause. An independent clause can stand as a sentence on its own while a dependent clause relies on another clause for full grammatical meaning. A complex sentence is usually made up of an independent clause joined by at least one dependent clauses. Therefore, one thing the second sentence of this introductory passage tells us is that her writing and grammatical style is generally complex.
You might also notice that it is actually pretty straight forward language. Neither figurative nor descriptive language can be found in these two sentences. In figurative language, the writer uses language in a non-traditional way in order to convey special meaning. Similes and metaphors are both types of figurative language. Descriptive language is the use of many adjectives or words that relate to the five senses in order to create vivid imagery. Rather than using figurative or descriptive language, Austen is simply using these two sentences to convey an argument. Therefore, one thing we know that is characteristic of her style is that it frequently lacks both figurative language and description.
Another thing you might notice is that Austen actually does not use any action verbs in these two sentences. Instead, she primarily uses auxiliary verbs, such as is and be. She also uses stative verbs, verbs that describe a state of mind or a situation. Examples are "acknowledged", "fixed", and "considered." All of the above describe mind sets and states of being, showing us that Austen's style primarily aims at moralizing and philosophizing rather than describing any action.