"Bunbury," or "Bunburyism," refers to the imaginary friend of Algernon called Bunbury that he uses to enable him to get out of awkward social engagements and to lead a double life. Bunbury finds his parallel in Ernest, Jack's equally imaginary brother, whose wayward exploits allows Jack to live both a respectable life of being an older, more responsible brother, but also he is able to be more reckless and live as he wants to in town. Bunburyism refers therefore to the double life that is such an important theme of the play, pointing towards the hypocrisy at the centre of Victorian society. Both Ernest and Bunbury give Jack and Algernon respectively the appearance of being much more virtuous and noble characters than they actually are. Note what Algernon says to Jack when Jack tells him that when he marries Gwendolen he will be able to kill off his brother:
Nothing will induce me to part with Bunbury, and if you ever get married, which seems to me extremely problematic, you will be very glad to know Bunbury. A man who marries without knowing Bunbury has a very tedious time of it.
This is typical humour from Wilde: Algernon suggests that, far from removing the need for a double life, marriage necessitates it, and that living in close proximity to somebody as in marriage makes it vital to have a double life in order for the marriage to work successfully. There is a rather dark humour to this announement of Algernon's, as on the one hand it clearly is a very funny comment, but on the other hand it points towards the widespread hypocrisy within Victorian society and highlights the reality behind marriage.