Explain how Jane Austen's novel Emma shows marriage and social classes in the nineteenth century.

This is a big question, but in brief, through her depiction of the village of Higbury as seen through Emma Woodhouse's eyes, Austen offers a sharp outline of class distinctions in Regency society and shows the importance of marriage to a woman's situation in life.
Emma is at the top of the social heap in her village, as her family owns property, has a good pedigree, and is wealthy. Emma herself is possessed of 10,000 pounds, a sizable fortune at the time, and lives with her father in a fine home with many servants and a carriage.
Her snobbery helps us understand social class in the village. She looks down on Mrs. and Miss Bates because, although of her same social class, they have become poor. They are poor because Mrs. Bates is widowed and Miss Bates never married. (They have no male to support them.) Their beautiful and accomplished niece Jane Fairfax, who Emma thinks of as "elegant," is also poor, and Emma pities her. Having social class is important, but so, as Emma understands, is money.
Emma, however, also looks down on characters who have money but lack her pedigree, such as the Coles, whose money comes from trade rather than landed wealth. She comically and snobbishly states she will refuse to go to their party—until suddenly she is the only one in the village not invited!
Emma also thinks she can advance her pretty but poor companion Harriet, who is an illegitimate child.
Emma grows during the novel to see that worth doesn't necessarily rely on social status or money, coming to be sorry, for example, that she has been mean to Miss Bates. She also has to acknowledge the limits of her own social power; for example, she is not able to marry Harriet to a man of means and status, and, by the end of the novel, gladly accepts Harriet's marriage to a mere farmer.
As for marriage, Jane Fairfax, despite her beauty, talents, and grace, faces a grim future because she has no dowry and, therefore, little chance of marrying. She has to face the humiliation of falling down the class ladder by becoming a governess, which she understands (more or less rightly) as a form of slavery. Although raised as a lady, she cannot hold on to that status once she accepts money for a job—something no lady does—and is therefore very fortunate that Frank Churchill wants to marry her. The difference between being the wealthy Mrs. Churchill and the poor governess Miss Fairfax is stark. Miss Taylor, Emma's former governess, also has a great rise in status when she marries Mr. Weston.
Likewise, although she declares she will never marry, Emma comes to see the importance of it as the eligible men in her village are swept up by others. When she fears losing Mr. Knightley, marriage suddenly becomes very important to her—we are led to believe she is in love with him, but it also a fact that without a husband she will remain stifled as the single daughter of a fussy hypochondriac.
Class determines how people treat other people and what their life prospects are in this novel. A woman's status is dependent to a very great extent of marriage, and it makes a huge difference in the quality of life of women like Miss Taylor and Jane Fairfax to make good marriages.