Dishonesty is the overarching theme of the whole play--after all, the title of the play is a specific reference. To be earnest means to be honest. Most of the characters in the play are dishonest--especially Jack and Algernon, and it is their dishonesty that creates the premise of the whole story.
Jack has been dishonest about telling his having a brother named Ernest who gets into all kinds of scrapes that require Jack to come to the city to solve. In reality, Jack is just trying to get a break from his life in the country; and while in the city, he tells everyone that he is the made-up person Ernest. Algernon on the other hand lies about having a very sickly friend named Bunbury who needs his attention in the countryside. Algernon uses this made-up person to escape his obligations in the city. Algernon catches on to what Jack is doing and calls him out on his "Bunburying"--but Bunburying is really just lying, and both men are pretty talented at crafting their crazy stories.
The lying becomes a comical crisis when two different women think are engaged to Ernest Worthing. Cecily from the city thinks she is dating him (it is really Jack) and Algernon has decided to take on the persona in the country and visits Jack's ward, Cecily in the country. They promptly fall in love. Both women are furious to discover they have been lied to, but it is all resolved nicely, and there is a happy ending for all.
All of the characters are defined by and affected by the deceptions they play on others. In the case of this very funny play it all ends up as it should, but Wilde is clearly satirizing the deceptive nature of humanity where things don't always end in such a tidy manner.