When we think of Romanticism in terms of literature, we typically think of the British Romantic poets like Wordsworth, Coleridge, Keats, and Shelley. These writers published their poetry in the late 18th and early 19th centuries. The defining characteristics of Romantic poetry include an emphasis on the imagination and individuality. Romantics were writing partly in response to the previous Age of Reason, so they also embraced the supernatural, the mysterious, the mystical. In the United States, a little later in the 19th century, writers like Hawthorne and Poe similarly wrote in modes that could take the reader beyond the literal, physical world that surrounded them. Hawthorne even wrote a definition of romanticism in a preface to his novel The House of the Seven Gables, in which he says the following:
[a romance]—while, as a work of art, it must rigidly subject itself to laws, and while it sins unpardonably so far as it may swerve aside from the truth of the human heart—has fairly a right to present that truth under circumstances, to a great extent, of the writer’s own choosing or creation.
Hawthorne is saying that romance, unlike what he calls "the novel," does not have to be strict in the way it portrays reality; instead, romance writers can have more license in terms of how the invent and describe the worlds in which their characters exist.
All of that said, Jane Austen's novels, while written in the same time period as the work of the famous British Romantic poets, focus on reality and contain no supernatural or fantastical elements. Austen's works are more commonly described as social novels, or even as social satires. Nothing that occurs in her novels, though, is mystical or mysterious. In fact, she depicts the sometimes harsh realities of love and marriage in the Edwardian Age. Pride and Prejudice, for example, portrays the Bennet family's (or at least Mrs. Bennet's) desperation to marry off five daughters and the tragic truth that Mr. Bennet must leave his property to an unpopular cousin because he has no sons. Elizabeth Bennet faces the tough truth that Mr. Darcy's family does not see her as his equal in terms of social status. In Persuasion, Austen writes of the somber love story of two older characters who eventually find themselves back to each other. These are all realistic narratives based on the social, cultural, and economic climate of Austen's England.
As another reviewer said, however, some of Austen's female protagonists, like Elizabeth Bennet, and Emma in the novel named for her, certainly have strong personalities and do exert their individual wills at times. That is probably the only real overlap, other than the time period, between Austen's work and that of Romantic poets like Shelley or Wordsworth.
Most scholarly opinion holds that Jane Austen, though writing during the Romantic era, was not actually a part of the Romantic movement herself. Nevertheless, there are some elements in her work which could be interpreted in the light of romanticism and its defining themes. One such element is individualism. There are striking examples of individualism that we see in a number of Austen's works. One only has to think of Elizabeth Bennet in Pride and Prejudice to see how an individual may assert her true self despite the artificial constraints imposed upon her by society. No one would seriously suggest that Austen's heroines are romantic in the same way as Byron's Manfred, but the affirmation of individuality against society's accepted mores and conventions is a suitably romantic theme nonetheless, although Austen expresses it somewhat differently.
The development of the self provides us with an additional point of contact between Jane Austen and the Romantics. Each of her novels illustrates the moral development of an individual self over time, and each undergoes a considerable degree of maturation. A common theme among romantic poets such as Wordsworth was the passage of childhood or youth to adulthood—"Tintern Abbey" and the "Ode: Intimations of Immortality" are two examples of this. Marianne Dashwood in Sense and Sensibility undergoes a similar transformation, eventually coming to a sober, more mature outlook on love.
Perhaps one of the reasons why Jane Austen has tended not to be associated with the Romantics is that different standards of maturation applied to men and women at the time. Women were not expected to develop morally or intellectually; instead, they were supposed to achieve society's ideals and expectations of how a woman should behave and remain in that state. That being the case, critics have tended to overlook the parallels between Austen's heroines and male characters in countless romantic poems and novels. As women were not expected to develop in the same way as men, it became difficult to incorporate their life experiences within the ambit of romanticism's core thematic elements.
How did Romanticism influence Jane Austen? Not much if at all. The author of Pride and Prejudice and Sense and Sensibility lived from 1775 to 1817. The Romantic Period of English literature is usually dated from 1798, with the publication of Wordsworth and Coleridge's Lyrical Ballads, to 1850. One biographer notes that
she didn't sell a novel until 1803, her first actual publication was in 1811, and all of the novels whose first drafts had been written before 1800 were revised by her after 1809 before they were published -- so that her most important period of literary activity was 1810-1817.
So Romanticism was too new a "movement" to have much of an influence on her writing. In a letter to her sister, Austen states that she has just finished reading a poem by Byron, so she was aware of the literature.
The term "Romantic" does not refer to love and romance, but to a ideal of looking to imagination as the greatest mental faculty; we not only see the world around us but we create it when we think back on it and imagine it. Nature was important to the Romantics as well. They were the first "back to nature" buffs and believed that the "noble savage" was the happiest of all people.