Situational irony occurs when the outcome of a series of actions or an event is incongruous with what we might expect to occur as a result of those actions/events. This kind of irony distinguishes reality from the perception of what we think "ought" to be.
In "The Gift of the Magi," irony occurs when we discover that Della's and Jim's gifts have been rendered useless through the self-sacrifice they endured in order to obtain those gifts. Each has given up his/her most treasured possession in order to purchase something that would accent the other's possession.
In other words, Della has cut off and sold her long, beautiful hair in order to afford a platinum fob for Jim's watch, and Jim has pawned off his precious watch in order to obtain tortoiseshell combs for Della's lovely hair. The act of giving--and the means through which it is made possible--is profound, and yet the story ultimately shocks us when we discover that there will be no satisfying exchange of gifts at the end but rather only a realization that what has been purchased at a great cost no longer has value.
In the story, Della and Jim each sacrifice the thing that means the most to them in order to give a gift to the other. Della sells her beautiful hair in order to buy Jim a platinum fob chain for his pocket watch, which is his most prized possession. Jim sells his watch to buy Della tortoise shell combs to wear in her hair. It is ironic that each gave up what was dearest to him/her out of love for the other, yet the sacrifice was pointless. Della could not wear the combs Jim bought for her; Jim could not use the watch fob Della bought for him.