When Algernon and Jack are debating the honor of reading the message engraved in another man's cigarette case, Algernon declares that "it is absurd to have a hard and fast rule about what one should read and what one shouldn't." He claims that a large portion of modern culture actually depends on reading things that one is not supposed to. In the same conversation, Jack declares that it is absurd for Algernon "to think that every aunt should be exactly like [Algernon's] aunt." Also, when Jack tells Algernon that his real name is Jack and not Ernest, Algernon declares that it "is perfectly absurd" to say such a thing.
Later, Algernon calls Jack "absurdly careless about sending out invitations," meaning that he does not send them enough. Jack also calls Algernon's fictitious invalid friend, Bunbury, a person with an "absurd name." Lady Bracknell becomes irritated with Algernon when he says that he must go to Bunbury's bedside again, and she declares that his "shilly-shallying"—by which she means whether he lives or dies—is "absurd." There are several more incidents where this word is used throughout the play, mostly by Algernon and Jack, and the word seems to be used synonymously with the word ridiculous or preposterous. When someone finds something ridiculous in the play, they call it absurd.
The irony with all of the examples I've listed is that the person using the word absurd is usually saying something absolutely absurd themselves! It isn't actually absurd for Jack to tell Algernon his real name; what's absurd is that he made one up to begin with. The absurd thing about Bunbury is that Algernon has invented a dying friend so that he can escape social occasions he would rather not attend; the name of the invented sufferer is hardly what is most absurd about the matter. Moreover, it is actually absurd for Lady Bracknell to suggest that a man "shilly-shallies" with the decision of whether to live or die, as if he had any say anyway!