In the beginning of the play, Jack and Algernon seem to have known each other as friends for some time. They speak to each other sarcastically and say things that suggest that they have known each other a while. For example, in Act I, Algernon asks if it is seven, or time to go to dinner. Jack says, "Oh! it always is nearly seven," referring to Algernon's continual need to eat. Algy explains that he is hungry and Jack says, "I never knew you when you weren't," (enotes.com, eText, pg. 25). This dialogue suggests that they have known each other for a long time, but how they met is never addressed. As far as their personal friendship is concerned, however, the two men seem to be slightly disrespectful towards one another due to their differences of opinion on life, marriage, and responsibility. Jack is more conservative than Algy is, for instance, and it tends to give way to "clever" remarks and off-handed put downs between them. In the end, the two men discover that they are brothers and the relationship seems to be solidified and more understandable at the same time.
The relationship between Jack and Algy is a humorous one. It is as if Algy were a pesky little gadfly in Jack's life. However, he seems to be a gadfly that Jack can't stay away from.
At the beginning of the play, Jack has called on Algy, so Jack takes the initiative to connect with him. But as the conversation progresses, Algy tries to wheedle information out of Jack and generally gets under his skin. He tries to find out where Jack lives in the country and who Cecily is. While this is happening, the men exchange banter and fight over food--showing distinctly immature reactions toward each other.
When Algy arrives at Jack's place in the role of Ernest, Jack is incensed. He says, "Nothing will induce me to take his hand. I think his coming down here disgraceful. He knows perfectly well why." He tells Algy that he will never be allowed to marry Cecily, and Algy responds by saying it's just as unlikely that he will marry Gwendolyn. Again, they descend into a food fight over the muffins. Jack tells Algy to leave, and he refuses.
In the final scene when Jack finds out that Algy is his little brother, their whole relationship makes perfect sense. Algy has been acting like an annoying little brother to Jack all along. Jack states, "Algy, you young scoundrel, you will have to treat me with more respect in the future." He tells Algy, with great irony, that he has never treated him like a brother in all his life. Algy says, "I did my best, however, though I was out of practice."
At the end, the two men shake hands, and Algy's role as the pesky little gadfly hovering around Jack is confirmed as his rightful place.