The play The Importance of Being Earnest, by Oscar Wilde, is a comedy of manners which intends to satirize the oddities that were resounding among members of the middle and upper classes of Victorian England. These oddities manifest themselves mostly in the behavior and demeanor of these social classes and are often quite ridiculous and useless. It is precisely those qualities which Wilde aims to target with this play.
Conflict is an essential part of a comedy of manners, because the situations that are present in them are often contradictory, ironic, and senseless. In The Importance of Being Earnest we find a myriad of conflicting situations which bring out those behaviors of the upper classes that Wilde wanted to mock.
The first conflict we see in Act I is social. Lady Bracknell seeks a husband for her daughter, Gwendolen, but only a rich man with a title will do. Although Jack Worthing clearly has money he lacks a "name", that is, a respectable family name to which Lady Bracknell can connect her daughter for social purposes. Since Jack is a foundling and grew up adopted, he is unable to locate a family member to be able to satisfy Lady Bracknell. This is when we see the ridiculous (but funny) retort that she gives Jack advising him to acquire a family:
I would strongly advise you, Mr. Worthing, to try and acquire some relations as soon as possible, and to make a definite effort to produce at any rate one parent, of either sex, before the season is quite over.
Therefore, until Jack finds some relations, Lady Bracknell will feel entitled to treat him badly since, in her mind, he is beneath her in social status. Therefore, he is not worthy of much. This is the actual conflict of class consciousness and class separation.
The second conflict is moral. Algernon lacks morals of any kind but, since he lacks them, does that make him immoral, or merely, amoral? His Bunburying is his way to escape responsibility from family, creditors, and events that he despises. Similarly, Jack creates this ill brother called Ernest whom he uses to also escape responsibility. The topic of double lives is everywhere in the play, and tells us about the hypocritical nature of the prudish Victorians. It is a conflict of truth versus lies; morality vs. immorality. It is just like Algernon tells Jack, when he expressed how foolish Jack would be for "getting rid" of his fake brother Ernest:
You don't seem to realize, that in married life three is company and two is none.
In all, The Importance of Being Earnest, presents conflict by exposing the ironic situations that occur in households where there are no real rules of decorum to be followed, and where everything is superficial. Wilde is effective in describing the concept of conflict under the perspective of the hypocritical and snobbish nature of the upper classes.