Another irony in the story "Hills Like White Elephants" is in the fact that this young man and woman have been traveling around together and presumably making passionate love, but then when the inevitable happens and the girl finds herself pregnant, the man doesn't want her to have the baby. The pleasure of love-making did not evolve for the sake of enjoyment but for making babies and bonding men and women together so that they could both provide protection and nurturing for the baby. The girl called Jig understands this, but the man--like many men--does not understand the connection between love and reproduction.
As a small addendum to that excellent response is the fact that they cannot mention--name--what in fact they are talking about. They do not mention the word "abortion," leaving a gap in the conversation that says more than the word itself ever could. The irony is that not saying the word carries more meaning to us, the audience, than if they did mention the word: absence is stronger than presence.
The irony of the story lies within the title. As the couple has their tense discussion of an unwanted pregnancy and a possible abortion, they overlook a hillside that the girl says reminds her of white elephants. This is ironic because white elephants are very rare and considered "precious." This also how the girl views her unborn child; she does not want the abortion (even though she orders a alcohol). Additionally, despite the beauty and rarity of a white elephants, they are also quite burdensome due to their size and insatiable appetite. The couple's baby also seems burdensome-- the girl risks losing her man if she does not have the abortion and she risks being unhappy if she does have the abortion.